Elliptic Diophantine Equations: A Concrete Approach via the Elliptic Logarithm 9783110281149, 9783110280913

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De Gruyter Series in Discrete Mathematics and Applications 2 Editor Colva M. Roney-Dougal, St Andrews, United Kingdom

Nikos Tzanakis

Elliptic Diophantine Equations A Concrete Approach via the Elliptic Logarithm

De Gruyter

Mathematics Subject Classification 2010: 11D25, 11D41, 11D88, 11G05, 11G07, 11G50, 11H06, 11J86, 11Y16, 11Y50, 11-04, 14E05, 14H52, 14Q05, 33E05, 52C07, 68W30 Author Nikos Tzanakis University of Crete Department of Mathematics Voutes Campus 70013 Heraklion, Crete Greece [email protected]

ISBN 978-3-11-028091-3 e-ISBN 978-3-11-028114-9 Set-ISBN 978-3-11-028115-6 ISSN 2195-5557 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P T P-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

To my beloved wife Maro, and to our family’s younger mathematicians, Eleni & Alexis, and Giorgos, with whom I share in life mathematics and so much more.

Preface

This book 17 0 is about elliptic Diophantine equations, the most standard instance of such an equation being the equation y 2 D x 3 C Ax C B in integers x, y, where A, B 2 Q and the right-hand side has no multiple roots. Many more Diophantine equations are elliptic Diophantine equations; in Chapter 1 the term is explained in its generality. More specifically, the main theme of this book is the explicit resolution of such equations and the resolution method that is exposed in detail is called elliptic logarithm method or, briefly, Ellog, in accordance with the terminology and notation introduced in [55]. The method has two main characteristics which are, first, the exploitation of the group structure with which the points of a non-singular cubic curve are endowed and, second, the use of linear forms in elliptic logarithms. The method owes its name to its second characteristic, the most modern. Immediately below we give more explanations. The first main characteristic (or ingredient) of the Ellog, which makes possible the transition from the elliptic Diophantine equation to linear forms in elliptic logarithms – the second main characteristic – is the fact that, on the one hand, an elliptic Diophantine equation can be transformed, by an appropriate “change of variables”, into a non-singular cubic equation of a special shape1 and, on the other hand, that the set of points2 of the curve defined by this cubic equation, is endowed with the structure of a finitely generated abelian group. The operation of “addition” in this group is a consequence of the simple observation that the third point of intersection with the curve of a line joining two points of the curve with coordinates in a number field also has coordinates in this same number field; if the two points coincide, as “the line joining them” we understand the tangent at the point. Thus, if we know two distinct rational, say, solutions (points), we can obtain a third one, also with rational coordinates, which is the third intersection with the curve of the line joining the two known points; and if we know only one solution (point), as a “line joining the two known points” we consider the tangent at the known point. This is the chord and tangent method for generating new solutions from known ones. At this point I refer the reader to the beautiful booklet of I. G. Bashmakova [3]. Bashmakova seems to believe that, in the two problems below, found in Arithmetica, Diophantus applied the chord and tangent method consciously. I am very cautious about this view; nevertheless, one might ex1 2

The Weierstrass equation. With coordinates rational or, more generally, in a number field.

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plain Diophantus’s solution to these problems as an application of this method, which is what I do below, following Bashmakova’s exposition [3, Chapter 6]. Problem Δ-24 of Diophantus’s Arithmetica [61, pp. 242–244].3 To divide a given number into two numbers such that their product is a cube minus its side. Original text: Δοθέντα ἀριθμὸν διελεῖν εἰς δύο ἀριθμούς, καὶ ποιεῖν τὸν ὑπ’ αὐτῶν κύβον παρὰ πλευράν. Sketch, in modern language, of Diophantus’s solution. As a “given number” he takes 6 and the two numbers in which 6 is “divided” are 6 and 6  x. Let y be the “side of the cube”, so that, by the problem, x.6  x/ D y 3  y. Diophantus puts y D ax  1 with a temporarily not specified.4 His first attempt, setting a D 2, is not successful because the coefficients of x are not cancelled out. Therefore he takes a D 3 and then his equation becomes 27x 3  26x 2 D 0, which gives the non-zero solution x D 26=27, y D 136=27. A deeper explanation of the above solution. Consider x.6  x/ D y 3  y as a curve possessing the obvious point .x, y/ D .0, 1/. The tangent to the curve at this point meets the curve at three points, two of them being .0, 1/ counted twice and the third will be the new sought-for point. The equation of the tangent is y D 3x  1 and we are led to Diophantus’s choice a D 3. In this solution, only one point, namely .0, 1/, was a priori known, so that the tangent was used. An analogous use of tangent explains the duplication formula of Bachet (1621), by which he was able to find rational solutions .x, y/ with xy ¤ 0 to the equation y 2 D x 3 C c for any integer c ¤ 1, 432, once he knew one rational solution .x1 , y1 / with x1 y1 ¤ 0; see [46, Introduction]. Problem Δ-26 of Diophantus’s Arithmetica [61, pp. 248–250].5 To find two numbers such that their product augmented by either gives a cube. Original text: Νὰ εὑρεθῶσι δύο ἀριθμοί, ὅπως τὸ γινόμενον αὐτῶν σὺν ἑκάτερον σχηματίζει κύβον. Sketch, in modern language, of Diophantus’s solution. As the first number he takes a multiple of a cube;6 specifically, 8x. He takes the second number equal to x 2  1. The conditions of the problem require that both 8x.x 2 1/C8x and 8x.x 2 1/Cx 2 1 be cubes. The first condition is satisfied by every x, while the second gives 8x 3 Cx 2 8x 3 4 5 6

Also [51, p. 199]. “I form a cube by arbitrary times minus its side” is my rough translation from Greek of Diophantus’s statement. Also [51, p. 203]. “I form the first by an arbitrary cube” is my rough translation from Greek of Diophantus’s statement.

Preface

ix

1 D y 3 . Again, Diophantus sets y D 2x  1, obtaining thus the solution x D 14=13; hence the sought-for numbers are 8  14=13 D 112=13 and .14=13/2  1 D 27=169. A deeper explanation of the above solution. Why the substitution y D ax  1 with a D 2? What is special about this value for a? In projective coordinates .X : Y : Z/ the above cubic equation becomes7 8X 3 CX 2 Z8XZ 2 Z 3 D Y 3 and the equation of the above line through the point .0, 1/ – which is the projective point .0 : 1 : 1/ – is Y D aX Z. On this line, the “point at infinity” is .1 : a : 0/, and the requirement that this be also a point on the cubic curve forces a D 2. Thus, the solution of Diophantus is the third point of intersection of the (projective) cubic curve with the (projective) line joining the points .0 : 1 : 1/ and .1 : 2 : 0/. The necessary theory and tools related to the above are discussed mainly in Chapter 1 and, also, in Chapter 2. The second characteristic (or ingredient) relevant to the method from which this book takes its name, consists in the fact that to each elliptic Diophantine equation one or more linear forms in elliptic logarithms are attached, and the computation of upper and lower bounds for them is a major part of the method. The necessary theory and tools for this are developed in Chapter 3. A first complete image of Ellog, in general, which results from the combination of the two ingredients, is given in Chapter 4. Specialising the application of Ellog to various classes of elliptic Diophantine problems results in Chapters 5, 6, 7, and 8. Each of these chapters leads up to a theorem furnishing an upper bound for the absolute value of the linear form L, involved in the Diophantine problem, in terms of a critical parameter M > 0. In Chapter 9, a major step is achieved due to a Theorem of S. David [12], namely, a lower bound for jLj (the same L as above), again in terms of M (the same M as above) is obtained. All quantities in both the upper and the lower bound of jLj, except for M , are explicit; moreover, as it will turn out, the lower bound runs faster to infinity with M than the upper bound and this fact clearly implies an explicit upper bound for M . Why is this important? At this point, it is not possible to explain in a few sentences, the meaning of M . For the present, the reader should consider that an explicit bound for M would reduce the resolution of the Diophantine problem to that of checking which lattice points in a hyper-cube of side 2M satisfy a certain condition. This is what I call in this book an effective resolution to the Diophantine problem. It is important to stress the fact that this checking can be performed in practice only if M is “very small”; this issue is discussed later in this preface. Four specific examples corresponding, respectively, to Chapters 5, 6, 7, and 8 are discussed, resulting in explicit very large upper bounds for M ; in all cases this is larger than 1040 . Unfortunately, this is a general fact: In all specific Diophantine problems, the effective upper bound for M is so large (something of the size of 1030 , say, would be very 7

On setting x D X=Z and y D Y =Z.

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“friendly”) that, in practice, the checking of lattice points in the hyper-cube, mentioned a few lines above, is impossible. Is it possible to reduce the upper bound of M to a manageable size (say of a few decades)? This would lead to an explicit resolution of the Diophantine problem. The answer is, in principle, positive due to a reduction method developed by B. M. M. de Weger [72], which is based on the LLL-basis reduction algorithm of Lenstra–Lenstra– Lovász [27]. Chapter 10 presents everything related to the reduction of the upper bound of M . The reduction method is then applied to the examples of Chapter 9, leading to upper bounds for M at most 17. This small upper bound leads then, very easily, to the complete explicit solution. The theoretical idea of the method8 goes back to Lang [25], who also explains it in [26]; its brief explanation is found in [45, Chapter IX.5, “Linear forms in elliptic logarithm”]. The discussion so far and the contents of this book confirm that from theory to practice a long way had to be covered; the method became practical only in 1994, after the work of R. J. Stroeker and the author [54], and, independently, the work of J. Gebel, A. Peth˝o and H. G. Zimmer [15]. These two 1994 papers would not have appeared if the work of N. Hirata-Kohno [17] and S. David [12], on lower bounds of linear forms in elliptic logarithms, had not been previously published. In a correspondence9 during 1991–1992, I asked S. David if he could make explicit the constants involved in the results of [17]; I am extremely grateful to him for his accepting my far from non-trivial “challenge”, which turned out to be a really heavy work [12] of more than 130 pages! For a clear and relatively short description of the practical “1994 method” I refer to [50, Chapter XIII]. Chapter 11 is special. In it, the resolution of the Weierstrass elliptic equation in S integers is discussed. What do we mean by S -integers? If S is a finite set of primes, an S -integer is, by definition, a rational number, with the property that the prime decomposition of its denominator allows only primes belonging to S ; in particular, every usual (rational) integer is an S -integer. One has to develop a theory of p-adic elliptic logarithms (with p a prime) for the points of an elliptic curve, in analogy with the theory of elliptic logarithms developed in Chapter 3; this is done in Section 11.1. In Section 11.2 linear forms in p-adic elliptic logarithms are introduced and the p-adic version of Ellog, is developed. This section is inspired by the papers of N. Smart [47] and Peth˝o–Zimmer–Gebel–Herrmann [36]; the second paper completes the project set up in the first paper. When [36] was published, no explicit lower bound for linear forms in p-adic elliptic logarithms – the analogue in the p-adic case of S. David’s theorem – existed, except for that in [39] which, however, is applicable only to elliptic curves of rank at most two.10 Therefore the authors of [36]

Without its “details” : Lower bounds for linear forms in elliptic logarithms, reduction process and, in general, all computational aspects. 9 Handwritten letters of the good old days! 10 This result is improved in [18], but still treats the case of two p-adic elliptic logarithms. 8

Preface

xi

turned to the recent, for those days, paper [16]. Very recently,11 N. H. Kohno released a valuable paper [19], in which “the p-adic analogue of S. David’s theorem” is proved. This is what is used in Chapter 11 instead of [16]. I am grateful to Noriko, who, meeting my desire, worked hard in order to provide me with her theorem before the date that I had to send my manuscript to the editors. The chapter includes a specific example with the primes p D 2, 3, 5, 7 involved. About the style of the book. To what extent should my exposition take for granted standard (more or less) material found in the literature? I had this speculation mainly concerning Section 1.2 of Chapter 1, Chapters 2, 3 and 9, Section 10.1 of Chapter 10, and Section 11.1 of Chapter 11. My decisions are detailed below. The basic theory of elliptic curves is so beautifully written in various text-books – few of them (only) are included in my bibliography –, that my hypothetical contribution could be described by  C 1! Therefore, Section 1.2 of Chapter 1 includes only the very basic facts that will be needed and gives references. For the theory of heights, in Chapter 2, only a moderate use of p-adic theory is required. I found that standard texts either include so much material that reference to them would disorientate (with respect to this book’s aim) my reader, or they adopt a point of view not very appropriate for the present book, as they build the relevant theory by considering extensions of Qp .12 Concerning Weierstrass equations over C and R and the Weierstrass }-function, treated in Chapter 3, I do the following. Since the basic general theory is so neatly written, for example, in [1, Chapter 1], I decided that I need not provide any explanation. However, specialisation of the general theory to Weierstrass equations with real coefficients is absolutely necessary in order to build a practical theory of elliptic logarithms. I decided to discuss this issue, rather in detail, guided by my personal taste. I adopted a very classical point of view, mainly based on the old (still in print) nice book [73] and the personal notes of N. Kritikos from A. Hurwitz’s 1916–1917 E. T. H. lectures on elliptic functions.13 The hard core of Chapter 9 is a special – very important though – case of Sinnou David’s Theorem. As I already mentioned, this is the result of his memoir [12] of more than 130 pages. The theorem, in the form appropriate for the needs of this book (Theorem 9.1.2), is stated only and aspects of its application in practice are discussed. For the applications of Chapter 10 the main tool is the reduction technique of B. M. M. de Weger [72], which is based on the LLL-algorithm [27]. The style of [72] is very appropriate for this book,14 but discusses many more applications than those 11 Actually,

when I was ready to send my manuscript to the editors! this book’s purposes, working with non-archimedean absolute values on (finite) extensions of Q is much more appropriate. 13 These notes in Greek [23], prepared by the late Dr. I. Ferentinou-Nikolakopoulou, circulated around 1980 in the Department of Mathematics of the University of Crete. 14 Is it accidental that B. M. M. de Weger and I had a congenial collaboration for years? 12 For

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needed here. Therefore, in Chapter 10, among other issues, I expose the reduction process focusing on the particular applications of the book. In Section 11.1 of Chapter 11, p-adic elliptic logarithms and their linear forms play the fundamental role. The theory on which the construction of such logarithms is based is described in Chapter IV of J. Silverman’s valuable book [45], though from a point of view somewhat more general than necessary for this book. What I decided to do was state only the absolutely necessary facts from Silverman’s exposition “translated” into a language appropriate for practical applications. As in the case with S. David’s Theorem in Chapter 11, the very recent and extremely important theorem of N. Hirata-Kohno, mentioned before, is only stated in the form appropriate for the needs of this book (Theorem 11.2.5). Although a main characteristic of the book is its use of computational methods, it is not a book on Computational Number Theory; issues such as – to mention only a few examples – the actual computation of Mordell–Weil bases, the search for rational points on elliptic curves up to a certain bound, the computation of canonical heights, various aspects of the implementation of the LLL-algorithm, and/or improvements of existing methods and algorithms, are beyond the scope of the book. To this “rule” I allowed three exceptions: In Chapter 3, first, I did not refrain from discussing in detail the actual computation of a fundamental pair of periods for a Weierstrass equation with real coefficients, an issue that fits very well in the framework of the chapter.15 Second, again in Chapter 3, I did not resist the temptation to describe the very clever algorithm of D. Zagier [74] for the computation of elliptic logarithms. Third, in Chapter 8, I present an algorithm of J. Coates related to the computation of the coefficients of Puiseux series. Suggestions for reading this book. The Diophantine problems treated in this book are classified to five classes: Weierstrass, quartic elliptic, simultaneous Pell, general elliptic, and Weierstrass in S -integers; let us use for these problems the symbols Pi , where i D 5, 6, 7, 8, 11, respectively, with this numbering justified by the chapter where the corresponding problem is mainly (but not exclusively) discussed. Since Ellog is applied in the most direct manner to P4 , I would suggest that the reader starts by understanding the resolution of this problem. In general, in order to understand the complete and explicit resolution of problem Pi , I suggest the following scheme:  



Read carefully Chapter 1. Make a first superficial reading of Chapter 2 to become acquainted with heights, so that you can read Section 2.6; if you already know about heights, go directly to Section 2.6. Pay attention to the content of Section 3.5. An understanding of the previous sections of Chapter 3 is necessary, with the exception of Section 3.4 which you will need only if you are interested in the actual computation of periods.

15 Besides,

a detailed treatment of this issue is not easily found in the literature.

Preface  





xiii

Comprehend the content of the short Chapter 4. Read carefully Chapter i. If i = 11 do not proceed to Theorem 11.2.6; instead, proceed to the following step. From Chapter 9 read carefully Sections 9.1 and 9.2. If i = 11 go back to Theorem 11.2.6 and complete your study of Chapter 11. END! If i is not 11, read that Section among 9.3, 9.4, 9.5 and 9.6 which corresponds to the chosen Pi . If i is different from 11, proceed to Chapter 10, read Section 10.1 and chose among the subsections of Section 10.2 the one that corresponds to the chosen Pi . END!

Software packages. My frequent reference to the software packages PARI (free), and MAPLE is because I happen to have been acquainted with them for years. Alternatively, for the applications of this book, one could turn to SAGE (free). As this was developed very recently – comparatively to the previously mentioned packages –, I had not the time to gain experience with it; this is the only reason why SAGE is not mentioned in my applications. MAGMA

Final acknowledgments. The materials of Chapters 4, 5, 8, 9 and 10 are mostly based on joint-papers with Roel Stroeker published between 1994 and 2003. It was a real pleasure to cooperate with Roel, noble friend and brave co-traveller in the long and adventurous but beautiful trip in the field of elliptic Diophantine equations. Around that same period, other people worked independently on various aspects of elliptic Diophantine equations, from a similar point of view; I have in mind mainly (in alphabetic order) J. Gebel, E. Herrmann, A. Peth˝o, N. P. Smart and H. G. Zimmer. We always had fruitful, and friendly communication; also their work was an inspiration source in writing Chapter 11. All serious computations in the examples of this book, besides their obvious debt to the software packages mentioned above, owe much, though indirectly, to people on whose work the routines that I have used are, more or less, based; let me mention (alphabetically) J. Cremona, M. van Hoeij, A. K. Lenstra, H. W. Lenstra, L. Lovász, J. Silverman, M. Stoll, B. M. M. de Weger, D. Zagier and many anonymous (to me, at least) heroes who are behind the algorithms’ implementation in various packages. Generally speaking, this book owes something to every author whose name appears in the bibliography; to some of them it owes much more, as becomes clear from the frequent references to their work. I also thank Y. Thomaidis who, shared with me his professional views about some issues of Diophantus’s Arithmetica. Warm thanks to De Gruyter for its continued collaboration and to D. Poulakis for inciting me to write this book and his warm encouragement. I am grateful to P. Voutier for his careful reading of Chapters 2 and 3. Of course, I am absolutely responsible for anything wrong that possibly escaped his attention.

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I am indebted to my wife Maro for her lifelong support, and for her warm encouragement and patience when I was writing this book; this has been a main factor for its completion! Heraklion, Crete, May 12, 2013

Nikos Tzanakis

Contents

Preface 1

2

vii

Elliptic curves and equations

1

1.1 A general overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Elliptic curves and the Mordell–Weil Theorem . . . . . . . . . . . . . . . . . .

5

Heights

9

2.1 Notations and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2 Absolute values in a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Heights: Absolute and logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 A formula for the absolute logarithmic height . . . . . . . . . . . . . . . . . . . 18 2.5 Heights of points on an elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 The canonical height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3

Weierstrass equations over C and R

29

3.1 The Weierstrass } function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The Weierstrass equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3

: E.C/ 7! C=ƒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Weierstrass equations with real coefficients . . . . . . . . . . . . . . . . . . . . . 3.4.1  > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2  < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Explicit expressions for the periods . . . . . . . . . . . . . . . . . . . . . 3.4.4 Computing !1 and !2 in practice . . . . . . . . . . . . . . . . . . . . . . . 3.5

36 38 40 41 44

: E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1 . . . . . . . . . . . . . . . . . . 47

4

The elliptic logarithm method

54

5

Linear form for the Weierstrass equation

57

6

Linear form for the quartic equation

60

7

Linear form for simultaneous Pell equations

69

xvi 8

Contents

Linear form for the general elliptic equation

78

8.1 A short Weierstrass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.2 Puiseux series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.3 Large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.4 The elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.5 Computing in practice B1 of Proposition 8.3.2 . . . . . . . . . . . . . . . . . . . 89 8.6 Computing in practice B2 and c9 of Proposition 8.4.2 . . . . . . . . . . . . . 91 8.7 The linear form L.P / and its upper bound . . . . . . . . . . . . . . . . . . . . . . 94 9

Bound for the coefficients of the linear form

98

9.1 Lower bound for linear forms in elliptic logarithms . . . . . . . . . . . . . . . 98 9.2 Computational remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.3 Weierstrass equation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.4 Quartic equation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.5 Simultaneous Pell equations example . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9.6 General elliptic equation: A quintic example . . . . . . . . . . . . . . . . . . . . 118 10 Reducing the bound obtained in Chapter 9

121

10.1 Reduction using the LLL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Weierstrass equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Quartic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 System of simultaneous Pell equations . . . . . . . . . . . . . . . . . . . 10.2.4 General elliptic equation: A quintic example . . . . . . . . . . . . . . 11 S-integer solutions of Weierstrass equations

125 125 127 131 134 137

11.1 The formal group of C and p-adic elliptic logarithms . . . . . . . . . . . . . 137 11.2 Points with coordinates in ZS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.3 The p-adic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 List of symbols

165

Bibliography

173

Index

177

Chapter 1

Elliptic curves and equations

1.1

A general overview

In this section we make an overview of general facts, terminology and conventions that will be used in this book. Let g.X , Y / be a non-zero polynomial with coefficients in a subfield K of C (in most cases, K D Q), irreducible over C, and let R be a subring of K (usually, but not always, R D Z) which will be fixed throughout this chapter. We are interested in solving the Diophantine equation g.u, v/ D 0,

.u, v/ 2 R  R.

(1.1)

The characterisation of the above equation as “Diophantine” comes from the requirement that the unknowns u, v belong to the prescribed ring R and not to the whole C or R. Solving the Diophantine equation is far different from solving the algebraic equation g.u, v/ D 0, in which the unknowns belong to C. The solutions of the algebraic equation define a curve or, more precisely, a model C of a curve; we state this by writing C : g D 0; g.X , Y / D a specific polynomial in X , Y and we say that C or, more precisely, the model C is defined by the polynomial g.X , Y /, or by the equation g D 0. Thus, we view C as the set C.C/ D ¹.u, v/ 2 C  C : g.u, v/ D 0º and the elements of C.C/ are called points of (the model) C . Sometimes we wish to focus our interest to the “real part” of C , which is the set C.R/ D ¹.u, v/ 2 R  R : g.u, v/ D 0º of real points of (the model) C . In general, if A is a subring of C, we set C.A/ D ¹.u, v/ 2 A  A : g.u, v/ D 0º and if .u, v/ 2 C.A/, we say that .u, v/ is an A-point of (on) C . The fact that the model C is defined by means of the polynomial g, whose coefficients belong to R, is expressed by saying that C is defined over R. Sometimes (actually very rarely) we will need to refer to the projective equation or, equivalently, to the projective model corresponding to equation (1.1). This results from the so-called homogenisation of the variables u and v, which consists in considering the equation g.U : V : W / D 0, def

(1.2) n

g.U : V : W / D W g.U=W , V =W /,

n D max¹degu g, degv gº.

Note that g.U , V , W / is homogeneous in U , V , W of degree n and g.u : v : 1/ D g.u, v/.

2

Chapter 1 Elliptic curves and equations

If g.U : V : W / D 0, then and only then g.kU : kV : kW / D 0 for every k 2 C  , therefore it is more appropriate to view the solutions of the equation g.U : V : W / D 0 projectively, i.e. as points .U : V : W / 2 P 2 .C/ rather than as solutions or affine points .U , V , W / 2 C 3 . If g.U : V : W / D 0 for some .U : V : W / 2 P 2 .C/ and there exists a k 2 C  such that kU , kV , kW 2 R, then .U : V : W / is a projective solution (point) over R. The dehomogenisation process from g D 0 to g D 0 consists in dividing g.U , V , W / D 0 through by W n and putting .U=W , V =W / D .u, v/. In the homogenisation process, from every solution .u, v/ of g D 0 we obtain a projective solution .u : v : 1/ of g D 0. In the dehomogenisation process, a solution .U : V : W / of g D 0 furnishes a solution .u, v/ of g D 0 if, and only if, W ¤ 0, the solution in this case being .u, v/ D .U=W , V =W /; but projective solutions of the form .U : V : 0/ cannot be “dehomogenised” to solutions .u, v/ of (1.1). Such solutions .U : V : 0/ are characterised as solutions (points) at infinity of the equation (1.1). Now we proceed to discussing the important fact that different equations may define the same curve. In order to make this more precise, we need first the following definition: Definition 1.1.1. For i D 1, 2, let gi .X , Y / be non-zero polynomials in CŒX , Y , irreducible over C and consider the models Ci : gi D 0. We say that a birational transformation exists between C1 and C2 or, equivalently, that the models C1 and C2 are birationally equivalent if, for .i , j / D .1, 2/, .2, 1/, rational functions Uij , Vij 2 C.X , Y / exist such that: for .i , j / as above, if .ui , vi / 2 Ci .C/ and we define .uj , vj / D .Uij .ui , vi /, Vij .ui , vi //, then .uj , vj / 2 Cj .C/ and .Uj i .uj , vj /, Vj i .uj , vj // D .ui , vi /. Actually, the above definition of birational equivalence, though satisfactory for the needs of this book, is not very precise: What about points .ui , vi / 2 Ci .C/ for which Uij .ui , vi / or Vij .ui , vi / is not defined (i.e. .ui , vi / is a zero of the denominator)? We overcome these difficulties if, for any model C : g D 0, we consider its function field C.C / (see a few lines below) and we define the notion of birational equivalence of two models C1 and C2 by means of their function fields C.C1 / and C.C2 /. The function field of the model C : g D 0 is, by definition, the field C., /, where  is transcendental over C and  is algebraic over C./, satisfying g., / D 0. Equivalently, C.C / can be defined as the quotient field of the integral domain CŒX , Y =I , where I is the ideal g.X , Y /CŒX , Y  of CŒX , Y . If two models C1 and C2 are birationally equivalent, then there exists an isomorphism of their function fields which fixes C. For a treatment of these issues, very appropriate for the theoretical background of this book, we refer the reader to §§3,4 of the classical book [66]. For an alternative, or complementary, exposition the interested reader can refer to [45, Chapters I, II.1, II.2].

Section 1.1 A general overview

3

Let us come back to Definition 1.1.1 and impose the further conditions that gi 2 RŒX , Y  for i D 1, 2 and the four rational functions Uij , Vij have coefficients in K. Then we say that C1 and C2 are birationally equivalent over K. If K D Q.R/, the quotient field of R, then, obviously, any R-point .ui , vi / of Ci is mapped by the birational transformation of Definition 1.1.1 to a Q.R/-point .uj , vj / of Cj . Clearly, birational equivalence of models of curves is an equivalence relation. The equivalence class of a model C will be called a curve, denoted by C. Thus, for this book, a curve is an equivalence class of a model and will be denoted by a capital calligraphic letter. If C1 , C2 are models of the same curve C, then C1 and C2 share a number of important properties, a very important one being that they have the same genus and we thus speak about the genus of the curve C rather than the genus of this or that model. The notion of genus is not easy to define but, fortunately, an in-depth knowledge of this notion is not absolutely necessary for the main purpose of this book, which is the practical resolution of elliptic Diophantine equations. Anyway, the interested reader can refer, for example, to [45, Chapter II.5], especially Theorem 5.4. A more classical approach is found in [66, Chapter VI, §5.3]. Another classical analytic point of view for genus, very much in the spirit of the present book, is found in [4, Chapter III, §21]; see especially Corollary 2. A “picturesque” geometrical notion of genus, easy for everybody, is presented in [38, Appendix to Chapter 1]. Although this approach to genus is not practical for the purposes of this book, we recommend the reader unaccustomed with this notion to have a look at it. In this book, we will deal with Elliptic Diophantine Equations. These are equations (1.1) defining a model C : g D 0 of an elliptic curve over K, which means that the curve has genus one and the corresponding projective equation (equivalently stated, the corresponding projective model) (1.2) has a solution (has a point) in P 2 .K/. In our applications, K D Q but, as a tool, we will sometimes need to consider elliptic curves defined over a number field K as well. Also, in our applications R D Z, except for Chapter 11, in which R D ZS , the ring of S -integers, where S is a finite set of primes; see the beginning of that chapter. When we solve an elliptic Diophantine equation, defined by an equation (1.1), we employ properties of the elliptic curve C, a model of which is C : g D 0. Let us put g D g1 and C D C1 ; the method of solution that we will apply requires working with one or two further models Ci : gi D 0 (i D 2, 3) of C, also defined over R, with any two of C1 , C2 , C3 birationally equivalent by means of transformations Uij , Vij (cf. Definition 1.1.1) having coefficients in R (in other words, the said models are birationally equivalent over Q.R/). These birational transformations will be fixed during the process of the resolution of the Diophantine equation g1 D 0. We will exclude all the finitely many (and easily computed) exceptional points of C1 .C/ at which at least one of the rational functions U1j , V1j is not defined, focusing on non-exceptional points of C1 .C/, i.e. on points which map to points on Cj for j ¤ 1 by means of the birational

4

Chapter 1 Elliptic curves and equations

map .u1 , v1 / ! .uj , vj / D .U1j .u1 , v1 /, V1j .u1 , v1 //  .uj , vj / .u1 , v1 / D .Uj 1 .u1 , v1 /, Vj 1 .u1 , v1 // For the convenience of our notation we will avoid the use of subscripts in curve notation and denote the various (two or three) birationally equivalent models that will be used during the resolution of an elliptic Diophantine equation by, for example, C : g.u, v/ D 0, E : f .x, y/ D 0, D : f1 .x1 , y1 / D 0 instead of Ci : gi D 0 (i D 1, 2, 3); note the use of different letters for the variables of different models. In accordance with the chosen letters, the birational transformation e.g. from C to E will then be denoted by .U, V/ and its inverse from E to C by .X , Y/, so that .u, v/ ! .x, y/ D .X .u, v/, Y.u, v// .u, v/ D .U.x, y/, V.x, y//

 .x, y/.

(1.3) (1.4)

If we denote by C the curve, models of which are C and E, then, in accordance with what was previously said, a point P of (on) C is “visualised” by both a pair .u0 , y0 / satisfying g.u0 , y0 / D 0 and a pair .x0 , y0 / satisfying f .x0 , y0 / D 0 (and analogously if a third model of C is used), where .u0 , y0 / and .x0 , y0 / are related by (1.3) and (1.4), with .u0 , v0 / in place of .u, v/ and .x0 , y0 / in place of .x, y/. It will be convenient to express this by writing P C D .u.P /, v.P // D .u0 , v0 / and P E D .x.P /, y.P // D .x0 , y0 /. To sum up: In general, if we have to solve a certain elliptic Diophantine equation g.u, v/ D 0, we consider the model C : g.u, v/ D 0 and the elliptic curve C corresponding to the model C , along with one or two further models E : f .x, y/ D 0, D : f1 .x1 , y1 / D 0 which will remain fixed in the process of the resolution of the Diophantine equation. We will consider fixed birational transformations between any pair of models C , E, D as in (1.3) and (1.4). We will say that “P is a point of (on) C” and will state this by writing P 2 C, if there exist points P C D .u.P /, v.P // 2 C.C/, P E D .x.P /, y.P // 2 E.C/ such that the rational functions U.u, v/ and V.u, v/ are defined at .u.P /, v.P // (in other words, the point .u.P /, v.P // of the model C is non-exceptional) and the pairs .u.P /, v.P // and .x.P /, y.P // are related by means of (1.3) and (1.4), with .u.P /, v.P // in place of .u, v/ and .x.P /, y.P // in place of .x, y/.1 Consequently, in accordance with these conventions/notations, we have P 2 E,

P E D .x.P /, y.P // 2 E.C/,

P C D .u.P /, v.P // 2 C.C/

P C D .U.P E /, V.P E // D .U.x.P /, y.P //, V.x.P /, y.P // P E D .X .P C /, Y.P C // D .X .u.P /, v.P //, Y.u.P /, v.P // 1

And analogously if it is necessary to make use of a third model D of C.

(1.5)

Section 1.2 Elliptic curves and the Mordell–Weil Theorem

5

Finally, if, as in Chapter 3, we use only one model, say E, then we will omit the superscript E . Moreover, and in accordance with this “rule”, the term “point P ” will mean “point of the model E” and we will simply write P instead of P E . We emphasise, once again, that a deep comprehension of the notion of genus is not absolutely necessary for the purposes of this book. From a practical point of view, the following fact is very important. Fact 1.1.2. If C : g.u, v/ D 0 is a model of genus one, where g has rational coefficients, then C is birationally equivalent over Q to a “short Weierstrass model” E : y 2 D x 3 C Ax C B with A, B 2 Q by means of a birational transformation (1.5), the coefficients of which are real algebraic numbers of degree at most min¹degu g, degv gº. These coefficients, as well as A and B, can be explicitly computed. A practical general algorithm of M. van Hoeij for computing the short Weierstrass model and the birational transformation, mentioned in Fact 1.1.2, is described in [22]; an implementation of this algorithm is included in the package algcurves of the computer algebra system MAPLE. Given a non-singular point .u0 , v0 / 2 C.Q/ and the coefficients of g as input, this algorithm returns the coefficients A, B of the equation of the short Weierstrass model and the coefficients of the birational transformation (1.3)–(1.4), which belong to Q.u0 , v0 /. Although this is not very explicit in [22], it can be verified by careful scrutiny of [20, §§ 1 and 2.1] and [21, §§ 1–3.1] on which the algorithm of [22] is based. The inclusion A, B 2 Q.u0 , v0 / can be further improved to A, B 2 Q: by an argument found on pages 93–95 of [6], a simple transformation .x, y/ 7! .t 2 x, t 3 y/, for a conveniently chosen t 2 Q, maps the equation y 2 D x 3 C Ax C B to an equivalent short Weierstrass equation with coefficients in Q. The above-mentioned algorithm, can be applied to quite “exotic” examples of models C : g.u, v/ D 0, like those appearing in [56]. If g has a “more ordinary” form, one can see more directly the necessary birational transformation between C and the short Weierstrass model, or even do the necessary computations “by hand”; see [10, Section 7.2].

1.2

Elliptic curves and the Mordell–Weil Theorem

In this section we focus exclusively on elliptic curves. We confine ourselves to stating the main facts of the basic (only) theory, with which we assume that the reader is accustomed. For these facts and the relevant theory one may refer to standard books on elliptic curves, such as [6, 24, 45, 46],2 or even from sections in some number

2

Indicated titles; only a few with which the author happens to be more acquainted.

6

Chapter 1 Elliptic curves and equations

theory textbooks devoted to elliptic curves, e.g. [7, Section 5.9], [32, Section 5.5]3 . In this chapter we will mostly refer to [45, Chapter III]. Firstly, any elliptic curve E over K has a model E : y 2 C a1 xy C a3 y D x 3 C a2 x 2 C a4 x C a6 ,

(1.6)

where a1 , : : : , a6 2 K. We call this equation a Weierstrass equation and the model E is a Weierstrass model. Weierstrass equations with a1 D a2 D a3 D 0 are called short Weierstrass equations and the corresponding models short Weierstrass models (cf. Fact 1.1.2). The first condition characterising E as an elliptic curve is that its genus be one (see page 3); equivalently, this is expressed by the condition E ¤ 0, where E is the discriminant of E.4 Also, we denote by E.K/ the projective model corresponding to C and observe that O D .0 : 1 : 0/ 2 E.K/ so that the second requirement for characterising E as an elliptic curve is also fulfilled. The point O is called the zero point of the curve for reasons that will become clear below; since O is not visualised on the affine plane, its traditional name, coming from Projective Geometry, is the point at infinity. The crucial fact about elliptic curves is that in E.K/ we can define by a very natural geometrical way, the so-called chord-tangent method, an operation C, called addition. Then, .E.K/, C/ is an abelian group with O its neutral element, which explains why we call it zero point; see [45, Section III.2], [46, Section I.2], [24, Section III.3], [7, Sections 5.9, 5.10], [6, Chapter 7], [32, Section 5.5]. The result of the addition of two points is their sum; the coordinates of the sum of two points are expressed as rational functions with coefficients in K of the coordinates of the two points; see [45, Group Law Algorithm 2.3], [24, Section III.4], [7, Section 5.10], [32, Definition 5.159]. The extremely important fact is that, when K is a number field, the group E.K/ is finitely generated, as L. J. Mordell [33] and A. Weil [71] proved (Mordell in the case K D Q and Weil for a general number field K); this is the famous Mordell–Weil Theorem: Theorem 1.2.1 (Mordell–Weil Theorem). Let K be a number field and consider the elliptic curve model E given by (1.6). Consider also the abelian group E.K/ and denote by Etors .K/ the torsion subgroup of E.K/. There exists a non-negative integer r, the rank of E over K, such that the following group isomorphism E.K/ Š Etors .K/  Zr , is valid. Proof. For the original proofs we refer to [33] and [71]. A proof of a very classical flavour in the case K D Q is found in [34, Chapter 16]. For a modern proof when K D Q we refer to [45, Chapter VIII], especially Section VIII.4. 3 4

Comment analogous to that of footnote 2. [45, pages 42–43].

Section 1.2 Elliptic curves and the Mordell–Weil Theorem

7

Practically, the Mordell–Weil Theorem means the following. If r  1, there exist points P1E , : : : , PrE in E.K/, such that: (1) PiE 2 E.K/ and PiE is of infinite order (i D 1, : : : , r). (2) For every point P E 2 E.K/, there exist integers m1 , : : : , mr and T E 2 Etors .K/, such that P E D m1 P1E C    C mr PrE C T E

(1.7)

and .m1 , : : : , mr , T E / is unique. (3) If r D 0, then E.K/ D Etors .K/. By another famous theorem due to B. Mazur (see [29, 30], or [45, Theorem 7.5]), Etors .Q/ is isomorphic as a group either to Z=N Z, with 1  N  12, N ¤ 11, or to Z=2Z  Z=2N Z with 1  N  4. Moreover, for every specific elliptic curve (1.6) defined over Q, Etors .Q/ is explicitly calculable by the Lutz–Nagell Theorem; see, for example, [24, Theorem 5.1] or [45, Corollary 7.2]. If r  1, then the union of the set of points ¹P1E , : : : , PrE º and a set of generators for Etors .K/ form a Mordell–Weil basis over K of E. The proof of Mordell–Weil Theorem is not constructive (even in the case K D Q), hence there is no guarantee that one can compute effectively a Mordell–Weil basis for any given E. However, very powerful techniques have been developed for elliptic curves over Q which are implemented in routines of standard software packages. The most standard reference about these techniques is John Cremona’s book [11], a true treasury of elliptic curves! Thus, although the explicit computation of a Mordell–Weil basis is not theoretically guaranteed, nevertheless, for “any reasonable” elliptic curve over Q, it is very likely that we can obtain such a basis by applying the relevant routines of software packages, like MAGMA, PARI, SAGE and others; see the item “software packages” on J. Cremona’s home page. Therefore, in this book we will not discuss further this issue and will take for granted that, when we solve an elliptic Diophantine equation with rational coefficients, a Mordell–Weil basis for the related elliptic curve is at our disposal. Addition of points in general elliptic models. We revisit Fact 1.1.2, keeping the same notation etc. Thus, we work now with two models C , defined by equation (1.1) with rational coefficients, and E, as in (1.6), again with rational coefficients, or as in Fact 1.1.2. We denote by E the elliptic curve, two models of which are C and E. The fact that we work simultaneously with two distinct models forces us to use superscripts on points (see Section 1.1). The transition from a point P C of C to the point P E of E and vice-versa is ruled by (1.5). Suppose that Q1 , Q2 are points on E. In view of the discussion in Chapter 1.1, this implies that, for i D 1, 2, there exist points PiE on E and points PiC on C , related by a birational transformation (1.4)–(1.3) and (1.5). Implicitly, it is understood that U, V are defined on P E and X , Y are defined on P C .

8

Chapter 1 Elliptic curves and equations

By the previous discussion we can form the sum Q1E C Q2E , which is a point, say E P , of the model E. If the birational transformation (1.4) is also defined on P E , then, in accordance with the conventions adopted in Section 1.1, P E is the representative of a point P of E. Then, this will allow us to write Q1 C Q2 D P , i.e. by definition, def

Q1 C Q2 D P ” Q1E C Q2E D P E . If, moreover, the birational transformation (1.5) is defined on Q1E , Q2E and P E , then the following makes sense: def

Q1C C Q2C D P C ” Q1E C Q2E D P E . Therefore, in view of these definitions along with (1.5), we see that .Q1 C Q2 /C D P C D .U.P E /, V.P E // D .U.Q1E C Q2E /, V.Q1E C Q2E //. The above definitions and conclusions are obviously generalised to an arbitrary (finite) number of points. More particularly, let Q1 , : : : , Qs be points on E and let m1 , : : : , ms be arbitrary integers. Then, def

m1 Q1 C    C ms Qs D P ” m1 Q1E C    C ms QsE D P E .

(1.8)

and .m1 Q1 C    C ms Qs /C D .U..m1 Q1E C    C ms QsE /, V..m1 Q1E C    C ms QsE //. (1.9) Finally, we give a simple useful lemma, that we will apply in subsequent chapters. Lemma 1.2.2. If the coefficients a1 , a2 , a3 , a4 , a6 in (1.6) are in Z and .x, y/ 2 E.Q/, then .x, y/ D .x1 =z 2 , y1 =z 3 /, where x1 , y1 , z 2 Z, z > 0 and gcd.x1 , z/ D 1 D gcd.y1 , z/. Proof. Let us write .x, y/ D .x1 =w1 , y1 =w2 /, where x1 , w1 , y1 , w2 are integers with w1 , w2 positive and gcd.x1 , w1 / D 1 D gcd.y1 , w2 /. Replacing for x and y in (1.6) we get w12 .y12 w1 C a1 x1 y1 w2 C a3 y1 w1 w2 / D w22 .x13 C a2 x12 w1 C a4 x1 w12 C a6 w13 /. We note that w12 is relatively prime to the number in the parenthesis in the right-hand side, therefore w1 jw2 and we write w2 D zw1 , where z is a positive integer relatively prime to x1 y1 . Replacing for w2 in the above equation we obtain w1 .y12 C a1 x1 y1 z C a3 y1 w1 z/ D z 2 .x13 C a2 x12 w1 C a4 x1 w12 C a6 w13 /. Now we observe that w1 is relatively prime to the number in the parenthesis in the right-hand side, hence w1 jz 2 , and, on the other hand, z 2 is relatively prime to the number in the parenthesis in the left-hand side, hence z 2 jw1 . It follows that w1 D z 2 and w2 D zw1 D z 3 , as claimed.

Chapter 2

Heights

In all Diophantine applications of the theory of elliptic curves we need a “measure” for the points of the elliptic curves involved in our investigations. This is accomplished with the notion of height of a point.1 In this chapter we develop the relevant theory from a point of view appropriate for the purposes of this book.

2.1

Notations and facts

We assume familiarity with standard algebraic number theory, as exposed, for example, in [35, Chapters 4 §3, 5, 6] and (partially) in [40, Chapter V]. For the convenience of the reader, we collect the basic facts that we will need without giving special reference to each of them. We will need to work with various relative finite extensions L=K=Q, where K and L may vary. The ring of integers of K will be denoted by ZK and analogously for L. For the elements of the number fields we will use small Greek letters. For the ideals of ZK we will use small fracture letters (like p, for example) and for the ideals of ZL we will use capital fracture letters (like P, for example). The symbol j between ideals means a divisibility relation, and is equivalent to the relation . If a is a non-zero ideal of ZK , then ZK =a is finite and we write NK=Q .a/ :D card.ZK =a/; this is the absolute norm of the ideal a. If ˛ 2 K  , then NK=Q .h˛i/ D jNK=Q .˛/j, where NK=Q .˛/ is the usual element norm of ˛. By default, the term ideal refers to integral ideal, i.e. ideal of the ring of integers. But we will also use fractional ideals, for which we will use fracture letters as well; the fact that an ideal is not necessarily integral will be emphasised by using the adjective “fractional”. If a is an ideal (integral or fractional) in K and we want to view it as an ideal in L, we will write aZL . If ˛ 2 K, we will write h˛i for the principal ideal ˛ZK . If we view ˛ as an element of L rather than as an element of K, then we will write ˛ZL for the principal ideal that ˛ generates in L. We will write P for the set of rational primes and PK , PL for the set of prime ideals of ZK and the set of prime ideals of ZL , respectively. If a is a non-zero fractional ideal in K and p 2 PK , we define the exponent of p in a and denoted by p .a/ as the exponent of p in the factorisation of a into prime ideals. If ˛ 2 K  , we will write p .˛/ instead of p .h˛i/. Analogous notations will be used for the non-zero ideals of L. 1

Actually, two notions of height of a point will be discussed.

10

Chapter 2 Heights

For P 2 PL there exists precisely one p 2 PK , such that PjpZK (the ideal P is over p or, in equivalent wording, p is below P). This p, in turn, is over a prime p 2 P, i.e. p is the unique rational prime divisible by p. With p, p and P related as just above, let us view p 2 PK as an ideal (not necessarily prime) of ZL , i.e. let us consider the ideal pZL of ZL and decompose it into prime ideals, as follows: Y PeL=K .P/ ; (2.1) pZL D Pjp

where eL=K .P/ is the ramification index of P relatively to L=K (hence eL=K .P/ D P .pZL ). The following relation is valid if P 2 PL is over p 2 PK : eL=Q .P/ D eL=K .P/  eK=Q .p/.

(2.2)

Further, the finite field ZL =P is a finite extension of the finite field ZK =p. The degree of this extension, which is called the degree of P relative to the extension L=K, is denoted by fL=K .P/. Similarly to (2.2) we have the relation fL=Q .P/ D fL=K .P/  fK=Q .p/.

(2.3)

Then the ideal norm of P relative to the extension L=K is, by definition, NL=K .P/ D pfL=K .P/ ;

(2.4)

note that, if in place of L and K we respectively have K and Q, this definition agrees with the one given above for the absolute norm. The definition of the ideal norm is extended by multiplicativity to all non-zero fractional ideals of L. Namely, if A is a non-zero fractional ideal in L and Y PP .A/ AD P2PL

is its prime decomposition, then, by definition, the ideal norm of A relative to the extension L=K is Y Y def pP .A/fL=K .P/ (2.5) NL=K .A/ D p2PK Pjp

and the norm of the zero ideal is defined to be the zero ideal. Two important properties of the norm are the following: The function NL=K from the group of fractional ideals of L to the group of fractional ideals of K is a homomorphism; moreover, for any fractional ideal A of L we have NL=Q .A/ D NK=Q .NL=K .A//.

(2.6)

If we have the decomposition (2.1), then, for every P dividing p we have the important relation X def dL=K .P/ D ŒL : K, dL=K .P/ D eL=K .P/  fL=K .P/; (2.7) Pjp

11

Section 2.2 Absolute values in a number field

dL=K .P/ is called local degree of L=K at P. Using (2.2), (2.3) and (2.7) we obtain another useful relation: X X dL=Q .P/ D eL=Q .P/fL=Q .P/ Pjp

Pjp

D eK=Q .p/eL=K .P/  fK=Q .p/fL=K .P/ X D eK=Q .p/fK=Q .p/ eL=K .P/fL=K .P/ D dK=Q .p/

X

Pjp

dL=K .P/ D ŒL : K  dK=Q .p/.

(2.8)

Pjp

2.2

Absolute values in a number field

Consider a number field K (K may coincide with Q). An absolute value on K is a function j  j : K ! R satisfying: (1) jxj  0 for every x 2 K, and jxj D 0 if and only if x D 0, (2) jxyj D jxjjyj, and (3) jx C yj  jxj C jyj, for all x and y in K. Two absolute values j  j1 and j  j2 are equivalent if there is a positive constant c 2 R such that jxj2 D jxjc1 for all x 2 K. This is equivalent to the fact that the metrics ıi .x, y/ D jx  yji on K induce equivalent topologies on K. An equivalence class of an absolute value on K is a place on K. All (different) places in K are obtained as follows: If p 2 PK and p is over p 2 P, then the place corresponding to p is represented by the absolute value j  jp which is defined by ´ p p .˛/=eK=Q .p/ D NK=Q .p/p .˛/=dK=Q .p/ if ˛ ¤ 0 . (2.9) j˛jp D 0 if ˛ D 0 This is a non-archimedean absolute value, which means (by definition) that it satisfies j˛ C ˇjp  max¹j˛jp , jˇjp º and, if j˛jp ¤ jˇjp , then strict inequality holds. In this way all non-archimedean places on K are obtained.2 The archimedean places on K are obtained as follows. Let ŒK : Q D d . There exist precisely d embeddings i : K ,! C (i D 1, : : : , d ). An embedding i : K ,! C is called real if  .K/  R, otherwise it is called complex. If s embeddings are real and

2

Easily then, every absolute value j  j equivalent to j  jp has the same property that characterises nonarchimedean absolute values: j˛ C ˇj  max¹j˛j, jˇjº and, if j˛j ¤ jˇj, then strict inequality holds. Therefore it makes sense to speak about a non-archimedean place.

12

Chapter 2 Heights

2t are complex, then d D s C 2t and we enumerate the embeddings as follows:  ,:::, „1 ƒ‚ …s real

sC1 , : : : , sCt , sCtC1 D sC1 , : : : , sC2t D sCt , „ ƒ‚ … complex

where  means complex-conjugate, so that  .x/ :D  .x/. o be the subset of EK consistLet EK be the set of all embeddings K ,! C and let EK ing of all pairwise non-conjugate embeddings; in particular card.EK / D s C 2t and o / D s C t. card.EK For each  2 E o we define the absolute value j˛j D j .˛/jd where in the right-hand side j  j denote the usual absolute value of C. These s C t absolute values are the archimedean absolute values; they are representatives of all possible archimedean places of K; an equivalent terminology is to say that these absolute values (places) are defined by the infinite primes of K. The product formula is the relation Y Y d .p/ j˛jp K=Q  j˛jd D 1, .˛ ¤ 0/. (2.10) o 2EK

p2PK

´ 1 if  is real d D 2 if  is complex. The proof of the product formula follows immediately from  Y  Y Y d .p/ j˛jp K=Q D NK=Q .p/p .˛/ D NK=Q pp .˛/ where

p2PK

p2PK

p2PK

D NK=Q .h˛i1 / D jNK=Q .˛/j1 and

Y

j˛jd D

o 2EK

Y

j .˛/j D jNK=Q .˛/j.

(2.11)

2EK

The following lemma is very useful, as it relates the absolute values in relative extensions. Lemma 2.2.1. Let L=K be an extension of number fields and p 2 PK . Then, (1) for any ˛ 2 K and any P dividing p, j˛jP D j˛jp , (2) for any ˛ 2 K,

ŒL:KdK=Q .p/

j˛jp

D

Y Pjp

d

j˛jPL=Q

.P/

.

13

Section 2.3 Heights: Absolute and logarithmic

Proof. Obviously, it suffices to consider a non-zero ˛ 2 K. With the aid of (2.2), (2.3) and the definitions of dL=Q .P/ and dK=Q .p/ we observe that dL=Q .P/ D eL=Q .P/fL=Q .P/ D eK=Q .p/eL=K .P/  fK=Q .p/fL=K .P/ D eK=Q .p/fK=Q .p/  eL=K .P/fL=K .P/ D dK=Q .p/  eL=K .P/fL=K .P/ which shows that eL=K .P/fL=K .P/ D

dL=Q .P/ . dK=Q .p/

We also note that, since the exponent of p in h˛i is p .˛/ and the exponent of P in p is eL=K .P/, it follows that the exponent of P in ˛ZL , i.e. P .˛/, is equal to p .˛/eL=K .P/. Using (2.4), the multiplicativity of the ideal norm and (2.6) we now calculate d

j˛jPL=Q

.P/

D NL=Q .P/P .˛/ D ¹NK=Q .NL=K .P//ºP .˛/ D ¹NK=Q .pfL=K .P/ /ºP .˛/ D NK=Q .p/fL=K .P/P .˛/ D NK=Q .p/fL=K .P/eL=K .P/p .˛/ D NK=Q .p/dL=Q .P/p .˛/=dK=Q .p/ d

D j˛jp L=Q It follows then that Y

d

.P/

j˛jPL=Q

.

.P/

P

D j˛jp

Pjp

dL=Q .P/

ŒL:KdK=Q .p/

D j˛jp

,

Pjp

where the right-most equality holds because of (2.8). The proof is now complete.

2.3

Heights: Absolute and logarithmic

The K-height of a projective point .x0 : x1 :    : xn / 2 P n .K/, where K is a number field, is defined as follows. First the finite-prime factor of the K-height is defined by Y max¹jx0 jp , jx1 jp j, : : : jxn jp ºdK=Q .p/ , HK,fin .x0 : x1 :    : xn / D p2PK

and the infinite-prime factor of the K-height is defined by Y HK,1 .x0 : x1 :    : xn / D max¹jxi jd º. o 2EK

i

Then, the K-height of the point is, by definition, HK .x0 : x1 :    : xn / D HK,fin .x0 : x1 :    : xn /  HK,1 .x0 : x1 :    : xn /. (2.12)

14

Chapter 2 Heights

Due to the product formula above, it is straightforward to check that, if ˛ 2 K  and .x0 : x1 :    : xn / 2 P n .K/, then HK .x0 : x1 :    : xn / D HK .˛x0 : ˛x1 :    : ˛xn /, which shows that the K-height of a projective point is independent from the choice of its projective coordinates. Moreover, due to the proposition below, for any number field K containing x0 , x1 , : : : , xn , the value of HK .x0 : x1 :    : xn /1=ŒK:Q is the same. Proposition 2.3.1. HK .x0 : x1 :    : xn /1=ŒK:Q is independent from the number field K containing the coordinates x0 , x1 , : : : , xn . Proof. We will show that, if L is a finite extension of K, then HL .x0 : x1 :    : xn /1=ŒL:Q D HK .x0 : x1 :    : xn /1=ŒK:Q . Actually, this is true for both the “finite-prime” and the “infinite-prime” factor separately. For the “finite-prime” factor it suffices to prove that HL,fin .x0 : x1 :    : xn / D HK,fin .x0 : x1 :    : xn /ŒL:K .

(2.13)

We calculate the left-hand side; to simplify notation, we write dP (respectively dp ) instead of dL=Q .P/ (respectively dK=Q .p/): HL,fin .x0 : x1 :    : xn / D

Y

max¹jxi jP ºdP D

P2PL

i

Y Y

max¹jxi jP ºdP .

p2PK Pjp

i

By (1) of Lemma 2.2.1, if Pjp, then jxi jP D jxi jp , therefore, in the right-most side of the above displayed equation, maxi ¹jxi jP º D maxi ¹jxi jp º D (say) jxi0 jp . Then HL,fin .x0 : x1 :    : xn / D

Y Y

dP

jxi0 jp º

P

D jxi0 jp

p2PK Pjp

D

 Y

Pjp

ŒL:K

dP

ŒL:Kdp

D jxi0 jp

(by (2.8))

max¹jxi jp ºdp

p2PK

D HK,fin .x0 : x1 :    : xn /ŒL:K as claimed. Next we prove that HL,1 .x0 : x1 :    : xn / D HK,1 .x0 : x1 :    : xn /ŒL:K .

(2.14)

15

Section 2.3 Heights: Absolute and logarithmic

First we note that every embedding  : K ,! C has exactly ŒL : K distinct extensions L ,! C. If is an extension of  we write j . Now we have Y Y max¹jxi j ºd D max¹ .xi /º HL,1 .x0 : x1 :    : xn / D o 2EL

D

i

Y Y

2EL

i

max¹j .xi /jº

2EK j

D

 Y

ŒL:K max¹j .xi /jº

2EK

D

 Y

ŒL:K max¹j .xi /jºd

o 2EK

D HK,1 .x0 : x1 :    : xn /ŒL:K . By (2.13), (2.14) and (2.12) we see that HL .x0 :    : xn / D HK .x0 :    : xn /ŒL:K , as required. Definition 2.3.1. Let .x0 : x1 :    : xn / 2 P n .K/, where K is a number field. Then, def

H.x0 : x1 :    : xn / D HK .x0 : x1 :    : xn /1=ŒK:Q is called the absolute height of the projective point .x0 : x1 :    : xn /. According to Proposition 2.3.1, this definition is independent from K. The absolute logarithmic height of .x0 : x1 :    : xn / 2 P n is, by definition, def

h.x0 : x1 :    : xn / D log H.x0 : x1 :    : xn /. The absolute logarithmic height of an algebraic number ˛ is, by definition, def

h.˛/ D h.1 : ˛/ D log H.1 : ˛/. The very basic properties of the absolute logarithmic height are stated in the following proposition: Proposition 2.3.2. If ˛, ˇ are algebraic numbers, then h.˛ˇ/  h.˛/ C h.ˇ/,

(2.15)

h.˛ C ˇ/  log 2 C h.˛/ C h.ˇ/

(2.16)

h.˛ n / D jnjh.˛/.

(2.17)

and, for every n 2 Z,

16

Chapter 2 Heights

Proof. Let K be any number field containing ˛ and ˇ. Denote by P the set of prime ideals of ZK and by EK the set of all embeddings K ,! C; note that card.EK / D ŒK : Q. For every p 2 P, let dp D dK=Q .p/. We have Y Y max¹1, j˛ˇjp ºd p  max¹1, j .˛ˇ/jº. H.˛ˇ/ŒK:Q D HK .˛ˇ/ D p2P

2EK

Using the inequality max¹1, xyº  max¹1, xº  max¹1, yº, valid for every pair of non-negative real numbers x, y, we obtain from the above displayed inequality H.˛ˇ/ŒK:Q  H.˛/ŒK:Q H.ˇ/ŒK:Q , from which the inequality (2.15) follows immediately on taking logarithms. In order to bound H.˛ C ˇ/ŒK:Q we first observe the following. If p 2 P, then max¹1, j˛ C ˇjp  max¹1, max¹j˛jp , jˇjp ºº  max¹1, j˛jp  max¹1, jˇjp . For  2 EK , max¹1, j .˛ C ˇ/jº  max¹1, j .˛/j C j .ˇ/jº  2  max¹1, j .˛/jº  max¹1, j .ˇ/jº. Note that, if in the last inequality we let  run through EK and we multiply the resulting relations, then the factor 2ŒK:Q will appear in the right-hand side. Using the above two displayed inequalities in HK,fin .˛ C ˇ/ and HK,1 .˛ C ˇ/, respectively, we get H.˛ C ˇ/ŒK:Q  HK,fin .˛/  HK,fin .ˇ/  2ŒK:Q HK,1 .˛/  HK,1 .ˇ/ D 2ŒK:Q HK .˛/HK .ˇ/ D .2H.˛/H.ˇ//ŒK:Q , which completes the proof of (2.16). Concerning (2.17), this is clear if n  0, so it suffices to prove that h.˛ 1 / D h.˛/. We use the identity max¹1, x 1 º D x 1 max¹1, xº, valid for every real x > 0. We have Y Y max¹1, j˛ 1 jp ºd p  max¹1, j .˛ 1 /jº H.˛ 1 /ŒK:Q D HK .˛ 1 / D D

Y

p2P

j˛

1

jp max¹1, j˛jp º



Y



p2P

D

Y p2P

dp

j˛ 1 jp

j˛ 1 j

Y

2EK

j .˛ 1 /j max¹1, j .˛/jº

2EK

 HK .˛/ D H.˛/ŒK:Q ,

2EK

where for the right-most equality we used the product formula (2.10). Thus H.˛/ D H.˛ 1 /, as required.

17

Section 2.3 Heights: Absolute and logarithmic

Heights over Q. Let us specialise our previous discussion to the important case of rational numbers. First, let us note that, for a point .x0 : x1 :    : xn / 2 P n .Q/, its absolute height is Y H.x0 : x1 :    : xn / D max¹jx0 j, jx1 j, : : : , jxn jº  max¹jx0 jp , jx1 jp , : : : , jxn jp º, p2P

where P is the set of (rational) primes. For ˛ 2 Q , j˛jp D p p .˛/ , p .˛/ is the exponent of p in the prime decomposition of ˛, and j0jp D 0. Now, let a0 , : : : , an 2 Z, not all zero, with gcd.a0 , : : : , an / D 1. Then, for every prime p, jai jp  1 for all i D 0, : : : , n and for at least one index i0 , ai0 is not divisible by p, so that jai0 jp D 1. Therefore, for every prime p, max¹ja0 jp , ja1 jp , : : : , jan jp º D 1. Consequently, H.a0 : a1 :    : an / D max¹ja0 j, ja1 j, : : : , jan jº. In general, for a0 , a1 , : : : , an 2 Z, not all zero, H.a0 : a1 :    : an / D

max¹ja0 j, ja1 j, : : : , jan jº . gcd.a0 , a1 , : : : , an /

(2.18)

For points of the form .1 : x1 :    : xn /, a useful expression is the following. If xi D ai =bi , where ai , bi are relatively prime integers (i D 1, : : : , n), and b D lcm.b1 , : : : , bn /  1, then H.1 : x1 :    : xn / D max¹b,

bja1 j bjan j ,:::, º D bmax¹1, jx1 j, : : : , jxn jº. (2.19) b1 bn

Indeed, .1 : x1 :    : xn / D .b : bja1 j=b1 : : : : : bjan j=bn /. Observe that gcd.b, bja1 j=b1 , : : : , bjan j=bn / D 1 and apply (2.18). For the last claim about the gcd it suffices to show that, for every prime p dividing b, some bai =bi is not divisible by p. For, if p k is the highest power of p dividing b (k  1), then p k is the highest power of p dividing bi for at least one i . But, since gcd.ai , bi / D 1, p does not divide ai ; hence bai =bi is an integer not divisible by p, as claimed. The absolute logarithmic height of a rational number q takes the form: X h.q/ D log H.1 : q/ D log max¹1, jqjº C log max¹1, jqjp º. (2.20) p

If q D a=b, where a, b 2 Z, we have, in view of (2.18), a second expression for h.q/ h.q/ D H.1 : a=b/ D H.b : a/ D log

max¹jaj, jbjº . gcd.a, b/

(2.21)

A third expression for h.q/ results as follows. Let q D a=b, where a, b 2 Z and write q as a fraction a1 =b1 in lowest terms. Then, specialising to n D 1 the right-most side of (2.19), we have H.1 : q/ D jb1 j  max¹1, jqjº and, since b1 D b= gcd.a, b/, we conclude that jbj . (2.22) h.q/ D log max¹1, jqjº C log gcd.a, b/

18

Chapter 2 Heights

2.4 A formula for the absolute logarithmic height This section aims at proving a very handy formula for the absolute logarithmic height. Our treatment is essentially that of D. Masser’s lectures [28], with slight modifications. We will need the notion of the height of a polynomial. Height of a polynomial. Let K be a number field and, as before, let PK be the set of prime ideals of the ring of integers ZK . The ring KŒX  of polynomials over K can be viewed also as the subset of K 1 (infinite-tuples) .ai /i0 , “almost all” (all but finitely many) of whose coordinates are equal to zero. Of course, the operations are defined by .ai / C .bi / D .ai C bi / and .ai /  .bi / D .ci /,

X

ci D

aj bk

.i  0/.

(2.23)

j CkDi

Let P D .ai / be a polynomial over K. The finite-prime height of P over K is, by definition, Y d max¹jai jp p º, HK,fin .P / D p2PK

where dp :D dK=Q .p/. We note that, for every ˛ 2 K, HK,fin .X  ˛/ D HK,fin .1 : ˛/. Also, if we view ˛ 2 K as the constant polynomial, then Y j˛jp D jNK=Q .˛/j1 , HK,fin .˛/ D

(2.24)

p2PK

where the right-most equality holds due to the product formula (2.10) and (2.11). The following proposition is, actually, a generalised and more sophisticated version of the so-called Gauss Lemma for Polynomials. Proposition 2.4.1. If P , Q 2 KŒX , then HK,fin .PQ/ D HK,fin .P /HK,fin .Q/. Proof. Obviously it suffices to prove our claim when both P and Q are non-zero polynomials. Let P D .ai /, Q D .bi / and PQ D .ci / (cf. (2.23)). Fix p 2 PK . Let P D .aj0 /, bk1 Q D .bk0 /. jaj0 jp D maxj ¹jaj jp º, jbk0 jp D maxk ¹jbk jp º and set aj1 0 0 Obviously, jaj0 jp  1 for every j and jbk0 jp  1 for every k, with equality holding for at least one subscript in both cases. P bk1 PQ D .ci0 /, where ci0 D j CkDi aj0 bk0 for every i , so that jci0 jp  1. Now aj1 0 0 Let j1 and k1 be respectively the least j for which jaj0 jp D 1 and the least k for P which jbk0 jp D 1. We have cj01 Ck1 D j CkDj1 Ck1 aj0 bk0 . Consider .j , k/ such that j C k D j1 C k1 . If j < j1 , then jaj0 jp < 1 and, since jbk0 jp  1, this implies

19

Section 2.4 A formula for the absolute logarithmic height

jaj0 bk0 jp < 1. If j > j1 , then k < k1 , so that jbk0 jp < 1 and jaj0 bk0 jp < 1. It follows P that, in ci0 D j CkDi ai0 bj0 , all summands but aj0 1 bk0 1 have p-absolute value strictly less than 1, while jaj0 1 bk0 1 jp D 1, leading to the conclusion that jcj01 Ck1 jp D 1. But, since jci0 jp  1 for every i , we conclude that maxi ¹jci0 jp º D 1 so that, d

d

d

d

d

max¹jci jp p º D jaj0 jp p  jbk0 jp p D max¹jai jp p º  max¹jbi jp p º. i

i

i

Since the above relation holds for every p 2 PK , the proof is now complete. Proposition 2.4.2. Let ˛ be an algebraic number and f˛ .X / D a0 X d C    C ad its minimal polynomial over Z. By this we mean that f˛ has integer coefficients, with a0 > 0 and gcd.a0 , : : : , ad / D 1; f˛ is irreducible over Q, and f˛ .˛/ D 0. Denote by ˛ .i/ , i D 1, : : : , d the roots of f˛ in C. Then,   Y d 1 .i/ max¹1, j˛ jº . h.˛/ D log a0 d iD1

Proof. We view ˛ as a complex number. Let K D Q.˛/, so that ŒK : Q D d , and Q let let L be the splitting field of f˛ over K, so that f˛ .X / D a0 diD1 .X  ˛i /, where ˛ D ˛ .1/ , say. Viewing a0 as a polynomial over L and applying (2.24) we see that HL,fin .a0 / D ja0 jŒL:Q . Since the coefficients of f˛ are relatively prime in Z, they are relatively prime also in L, therefore maxi ¹jai jP º D 1 for every prime ideal P of ZL ; consequently, HL,fin .f˛ / D 1. Also, since all fields Q.˛ .i/ / are isomorphic to Q.˛/, it follows that HL,fin .X  ˛i / D HL,fin .X  ˛/. Now we apply Proposition 2.4.1: Y HL,fin .X  ˛i / D ja0 jŒL:Q HL,fin .X  ˛/d 1 D HL,fin .f˛ / D HL,fin .a0 /  i

which shows that HL,fin .1 : ˛/d D HL,fin .X  ˛/d D ja0 jŒL:Q . By Proposition 2.3.1, HL,fin .1 : ˛/ D HK,fin .1 : ˛/ŒL:K , hence ja0 jŒL:Q D HL,fin .1 : ˛/d D HK,fin .1 : ˛/ŒL:Kd D HK,fin .1 : ˛/ŒL:Q , hence HK,fin .1 : ˛/ D ja0 j. By definition, h.˛/ D log H.1 : ˛/ D D

1 1 log HK .1 : ˛/ D log.HK,fin .˛/  HK,1 .˛// d d

1 log.a0  HK,1 .˛//. d

(2.25)

20

Chapter 2 Heights

It remains to compute HK,1 .˛/. There are exactly d embeddings i : K ,! C (i D 1, : : : , d ) and they are characterised by their image of ˛, namely,  .˛/ D ˛ .i/ . Q Therefore, HK,1 .˛/ D diD1 max¹1, j˛ .i/ jº. By inserting this into (2.25) we complete the proof.

2.5 Heights of points on an elliptic curve As always in this chapter, H. :    : / and h./ denote absolute height and absolute logarithmic height, respectively. Consider an elliptic curve E and its short Weierstrass model E : y 2 D x 3 C Ax C B,

A, B 2 Q.

(2.26)

Use will be made also of the following slightly different model of E D : y 02 D 4x 03  g2 x 0  g3 ,

g2 D 4A, g3 D 4B,

.x 0 , y 0 / D .x, 2y/.

For points P of E, such that P E 2 E.Q/, the naive or Weil height of P is, by definition, def h.P / D h.x.P //. In the special important case that P E 2 E.Q/, we put P E D .x, y/, so that P D D .x 0 , y 0 / D .x, 2y/, with x D a=b, a, b 2 Z, gcd.a, b/ D 1. Then, X def log max¹1, jxjp º D log max¹jaj, jbjº. (2.27) h.P / D h.x/ D log max¹1, jxjº C p

For S. David, whose main result in [12] is extremely important for the application of Ellog, it is more convenient to use a somewhat different height of P , namely, hD .P / D h.1 : x 0 : y 0 / D h.1 : x : 2y/. def

In the special case that E P D .x, y/ 2 E.Q/, hD .P / D h.1 : x 0 : y 0 / D h.1 : x : 2y/ X D log max¹1, jxj, j2yjº C log max¹1, jxjp , j2yjp º.

(2.28)

p

From the point of view of Diophantine equations, it is more convenient to work with the model E than with the model D, this last model being more appropriate for [12]. Therefore we need to see how the two heights are related. This we do in the following lemma. Lemma 2.5.1. Let E as in (2.26) and let P E D .x, y/ 2 E.K/, where K is a number field. Then 5 1 hD .P /  h.P / C 2 log 2 C .h.A/ C h.B//. 2 2

21

Section 2.5 Heights of points on an elliptic curve

Proof. We will make extensive use of Proposition 2.3.2, without special reference each time we use any of the relations (2.15), (2.16) or (2.17). Moreover, we will make use of the fact that, if x1 , x2 are algebraic numbers, then h.1 : x1 : x2 /  h.x1 / C h.x2 /

(2.29)

Indeed, if K is a number field containing both x1 and x2 , then it suffices to prove that H.1 : x1 : x2 /  H.1 : x1 /H.1 : x2 / which, in turn, amounts in proving that HK .1 : x1 : x2 /  HK .1 : x1 /HK .1 : x2 /. The proof is left to the reader, as it is very similar to the proof of (2.15). Now, we have 2h.y/ D h.y 2 / D h.x 3 C Ax C B/  log 2 C h.x 3 C Ax/ C h.B/ D log 2 C .h.x/ C h.x 2 C A// C h.B/ D h.x 2 C A/ C h.x/ C h.B/ C log 2 D .log 2 C 2h.x/ C h.A// C h.x/ C h.B/ C log 2 D 3h.x/ C h.A/ C h.B/ C 2 log 2. Using this and (2.29) we have, hD .P / D h.1 : x : 2y/  h.x/ C h.2y/  h.x/ C h.2/ C h.y/ 5 1 D h.x/ C h.y/ C log 2  h.x/ C .h.A/ C h.B// C 2 log 2. 2 2 Since in our future study we will make use almost exclusively of points with rational coordinates, it is worthwhile to give a more accurate relation between hD .P / and h.P /, when P E 2 E.Q/. This will be used in the proof of Proposition 2.6.1. Lemma 2.5.2. Let E be the short Weierstrass model (2.26). There exist constants c1 , c2 , depending only on E, such that, for any point P 2 E with P E 2 E.Q/, we have 3 c1 C h.P /  hD .P /  c2 C 32 h.P /. 2 Proof. Let P E D .x, y/ 2 E.Q/, so that P D D .x 0 , y 0 / D .x, 2y/ 2 D.Q/. In analogy with (2.28) we define (only for the needs of this proof) X def hE .P / D h.1 : x : y/ D log max¹1, jxj, jyjº C log max¹1, jxjp , jyjp º. p

We observe that, for every prime p, 1 max¹1, jxjp , jyjp º  max¹1, jxjp , j2yjp º 2  max¹1, jxjp , jyjp º < 2 max¹1, jxjp , jyjp º

22

Chapter 2 Heights

and 1 max¹1, jxj, jyjº < max¹1, jxj, jyjº  max¹1, jxj, j2yjº  2 max¹1, jxj, jyjº, 2 from which it follows that  log 2 C hE .P /  hD .P /  log 2 C hE .P /.

(2.30)

Now we will compare hE .P / with h.P / (cf. (2.27)). Firstly, we will prove that there exist positive constants C1 , C2 , depending only on E and not on P , such that C1  max¹1, jxjº3=2  max¹1, jxj, jyjº  C2  max¹1, jxjº3=2 .

(2.31)

Here is the proof for the right-most inequality (2.31). We have y 2 D x 3 C Ax C B. 2 3 If jxj  1, then y 2  1 C jAj C jBj; and pif jxj > 1, then y < .1 C jAj C jBj/jxj , 3=2 therefore, in both p cases, max¹1, jyjº  1 C jAj C jBj  max¹1, jxjº . Obviously, max¹1, jxjº  1 C jAj C jBj  max¹1, jxjº3=2 , therefore, max¹1, jxj, jyjº D max¹max¹1, jxjº, max¹1, jyjºº  C2  max¹1, jxjº3=2 , p where C2 D 1 C jAj C jBj. Now the proof of the left-most inequality (2.31). There exists a sufficiently large M > 1 such that, if jt j  M then jt 3 C At C Bj > 14 jt j3 . If jxj  1 then max¹1, jyjº  1 D max¹1, jxjº D max¹1, jxjº3=2 . If 1 < jxj < M , then M 3=2 max¹1, jyjº  M 3=2 > jxj3=2 D max¹1, jxjº3=2 , hence max¹1, jyjº > M 3=2 max¹1, jxjº3=2 . If jxj  M , then jyj  12 jxj3=2 , hence max¹1, jyjº  jyj  12 jxj3=2 D 12 max¹1, jxjº3=2 . Therefore, max¹1, jxj, jyjº  max¹1, jyjº  C1 max¹1, jxjº3=2 , where C1 D min¹ 12 , M 3=2 º. Next, we will compute a lower and an upper bound for hE .P /, both of which are of the form “constant C 32 h.x/”. These bounds will be combined with (2.30) to complete the proof of the lemma. For this purpose, let d be the least positive integer such that both d 4 A and d 6 B are integers. Obviously, d is a divisor of the denominators of A and B, which implies that d is independent from the point P D .x, y/. Now, y 2 D x 3 CAx CB implies that .d 3 y/2 D .d 2 x/3 C.d 4 A/.d 2 x/C.d 6 B/, hence .d 2 x, d 3 y/ satisfies a short Weierstrass equation with integer coefficients. By Lemma 1.2.2, it follows that d 2 x D r=t 2 and d 3 y D s=t 3 , where r, s, t are integers, t > 0 and gcd.r, t / D gcd.s, t / D 1. We have hE .P / D h.x, y/ D log H.1 : x : y/ D log H.1 : r=.dt /2 : s=.dt /3 / D log H..dt /3 : r.dt / : s/.

(2.32)

23

Section 2.6 The canonical height

By (2.18), H..dt /3 : r.dt / : s/ D

max¹.dt /3 , dt jrj, jsjº .dt /3 D  max¹1, jxj, jyjº, ı ı

(2.33)

where ı :D gcd..dt /3 , dt r, s/  gcd.d 3 t 3 , d 3 t r, d 3 s/  d 3 gcd.t 3 , t r, s/ D d 3 and we have used the fact that gcd.r, t / D 1 D gcd.s, t /. By (2.33) and (2.31), C1

.dt /3 .dt /3  max¹1, jxjº3=2  H..dt /3 : r.dt / : s/  C2  max¹1, jxjº3=2 . (2.34) ı ı

Applying (2.22) with q D x D r=.dt /2 , we get h.x/ D log max¹1, jxjº C log

d 2t 2 d 2t 2 D log max¹1, jxjº C log , gcd.r, d 2 t 2 / gcd.r, d 2 /

hence max¹1, jxjº3=2 D

gcd.r, d 2 /3=2  exp. 32 h.x//. .dt /3

Then, (2.34) becomes gcd.r, d 2 /3=2 3 h.x/ gcd.r, d 2 /3=2 3 h.x/  e2  H..dt /3 : r.dt / : s/  C2   e2 ı ı where, obviously, gcd.r, d 2 /3=2 1   d 3. d3 ı Combining the last two displayed relations with (2.32) we get C1 

log C1  3 log d C 32 h.x/  hE .P /  log C2 C 3 log d C 32 h.x/. Finally, combining the above relation with (2.30) and remembering that, by definition, h.x/ D h.P / (cf. (2.27)), we obtain the relation in the announcement of the lemma, with c1 D  log 2 C log C1  3 log d and c2 D log 2 C log C2 C 3 log d .

2.6

The canonical height

Let now E be an elliptic curve over a number field K and P 2 E. By Fact 1.1.2 of Section 1.1 there exists a Weierstrass model C :

y 2 C a1 xy C a3 y D x 3 C a2 x 2 C a4 x C a6

with coefficients in K. Working with the model C of E and its function field K.C /, we

24

Chapter 2 Heights

have the degree-two function x 2 K.C /, defined by C.K/ 3 P C D .x.P /, y.P // 7! x.P / 2 K and this function is even (x.P / D x.P /). Therefore, if P 2 E is such that P C 2 C.K/, then according to [45, pages 247–248], the canonical height of P C can be defined by the following limit: h.x.2N P C // 1 , lim 2 N !1 4N

(2.35)

where h./ denotes the absolute logarithmic height. This limit is independent from the Weierstrass model. Indeed, let C 0 : W .x 0 , y 0 / D 0 be another Weierstrass model over K of E. By [45, Proposition 3.1(b), Chapter III.3], x 0 .P / D 2 x.P /C for convenient

,  2 K. Working now with the model C 0 of E and its function field K.C 0 /, we have 0 the degree-two function x 0 2 K.C 0 /, defined by C.K/ 3 P C D .x 0 .P /, y 0 .P // 7! x 0 .P / 2 K and this function is even (x 0 .P / D x 0 .P /). Therefore, as before, the 0 canonical height of P C can be defined by 0

1 h.x 0 .2N P C // . lim 2 N !1 4N But x 0 is an even, degree-two function of K.C / also, in view of the simple relation between x 0 .P / and x.P /, therefore, by [45, Proposition 9.1, Chapter VIII.9], the two limits displayed above are equal and their common value (2.35) is defined as the canonO /, without any model ical or Néron–Tate height of the point P 2 E, denoted by h.P indication on P . Thus, taking also into account (2.27), we have N C O / D 1 lim h.2 P / . h.P (2.36) 2 N !1 4N It is worthwhile to note, although we will not need it in this book, that for the definition of the canonical height it is not necessary to use the function x 2 E.K/, but any even function in E.Q/; see [45, VIII.9]. As already noted, for S. David it is more convenient to work in [12] with models D like the one on page 20 and adopt hD as the absolute logarithmic height of a point, as defined in (2.28). As a consequence, in the notation just before the relation (2.28), the canonical height used by S. David is defined by

hD .2N P / . (2.37) N !C1 4N As an immediate consequence of the above relation and Lemma 2.5.2 we have proved the following proposition: hOD .P / D

lim

Proposition 2.6.1. Let E : y 2 D x 3 C Ax C B be a model of an elliptic curve E, where A, B 2 Q, and let D : y 02 D 4x 03  g2 x 0  g3 ,

g2 D 4A, g3 D 4B,

25

Section 2.6 The canonical height

O / so that D is also a model of E. Then, for any point P 2 E, the canonical heights h.P O and hD .P /, defined by (2.36) and (2.37), respectively, are related by O /. hOD .P / D 3h.P A very important property of the canonical height is that, by means of it, a positivedefinite quadratic form is defined as follows: First, one defines the so-called Néron– Tate (or Weil) pairing by O C Q/  h.P O /  h.Q/. O hP , Qi D h.P The following important properties for the canonical height and the Néron–Tate pairing hold (see [45, Theorem 9.3]):  



The Néron–Tate pairing is bilinear. O /; in particular, O For any P 2 E with P E 2 E.Q/ and any m 2 Z, h.mP / D m2 h.P O O h.P / D h.P /. O /  0 and h.P O / D 0 if and only if P E is a torsion point. h.P

Using these properties, it is not difficult to see that, if P E is expressed as in (1.7), then X O /D 1 h.P hPi , Pj imi mj . (2.38) 2 1i,j r

The function hO can be extended to the r-dimensional real vector space E.Q/ ˝ R, where E.Q/=Etors .Q/ sits as an r-dimensional lattice, and defines a positive definite quadratic form on this vector space; see [45, items 9.4-9.7]. Then, the height pairing matrix of .P1 , : : : , Pr / is, by definition, the positive definite matrix H D H.P1 , : : : , Pr / D . 12 hPi , Pj i/rr

(2.39)

(cf. (2.38)). The following result is crucial for the applications of this book: Proposition 2.6.2. Let P E be expressed as in (1.7). Then O /  max m2 , h.P i 1ir

where is the least eigenvalue of the height pairing matrix H defined in (2.39). Proof. According to (2.38) we have O / D mT Hm, h.P where m is the column vector with components m1 , : : : , mr . As H is symmetric, a def diagonal matrix ƒ of eigenvalues 0 < D 1 < 2 <    < r of H and an

26

Chapter 2 Heights

orthogonal matrix Q exist such that H D QT ƒQ. Writing n D Qm and observing that QT Q D Ir (identity matrix), we deduce O / D mT QT ƒQm D nT ƒn D h.P  D

r X iD1 r X iD1

r X

i n2i

iD1

n2i D  nT n D  mT QT Qm D  mT m m2i  c1 max m2i , 1ir

as claimed. Proposition 2.6.3. Let E : y 2 D x 3 C Ax C B with A, B 2 Q be an elliptic curve model. Let D : y12 C a1 x1 y1 C a3 y1 D x13 C a2 x12 C a4 x1 C a6 ,

(2.40)

be any Weierstrass model of the same elliptic curve, with a1 , a2 , a3 , a4 , a6 rational integers, such that the equation of D is related to the equation of E by a change of variables x1 D 2 x C , y1 D 3 y C 2 x C , for appropriate rational numbers , , , .3 Let D

1 1 1 1 log jj C logC jj j C logC jb2 =12j C log 2 12 12 2 2 1 C .log 2 C h.// C h. / C log j j C 1.07, 2

(2.41)

where  and j are, respectively, the discriminant and j -invariant of the model E, b2 D a12 C 4a2 , 2 D 1 or 2 according to whether b2 vanishes or not, respectively, and logC is defined for any real ˛ > 0 by logC ˛ D log max¹1, ˛º. Then, for every P E D .x.P /, y.P // 2 E.Q/, we have O /  1 h.x.P //   . h.P 2

(2.42)

Proof. Note that the two models D and E have equal j -invariants, while their corresponding discriminants 1 and  are related by 1 D 12 .

(2.43)

Now we apply Silverman’s Theorem 1.1 in [44] to the model (2.40). That theorem requires that the ai ’s be algebraic integers. In our proposition, these coefficients are 3

One can easily see that there always exists such a model D.

27

Section 2.6 The canonical height

rational integers, which permits to replace h./ and h1 .j /, h1 .b2 =12/, which appear in Silverman’s theorem, by log jj and logC jj j, logC jb2 =12j, respectively. We thus obtain the following inequality O /  1 h.x1 .P //   0 h.P 2

(2.44)

with 0 D

1 12

log j1 j C

1 12

logC jj1 j C 12 logC jb2 =12j C 12 log 2 C 1.07,

where 1 and j1 are, respectively, the discriminant and the j -invariant of the model D. As already noted, (2.43) holds and j1 D j , the j -invariant of E. Therefore, 0 D

1 12

log jj C log j j C

1 12

logC jj j C 12 logC jb2 =12j C 12 log 2 C 1.07. (2.45)

Now, h.x1 .P // D h. 2  x.P / C /  log 2 C 2h. / C h./ C h.x.P //

(by Proposition 2.3.2),

therefore, O /  1 h.x.P //  h.P O /  1 h.x1 .P // C 1 .log 2 C h.// C h. / h.P 2 2 2 1 0 (by (2.44)).   C .log 2 C h.// C h. / 2 On combining the last inequality with (2.45) we obtain the inequality (2.42). Remark. According to the announcement of Proposition 2.6.3, the Weierstrass model D must fulfil certain conditions, but otherwise it is arbitrary. What we do in practice when we have to solve a specific Diophantine equation, is to choose (2.40) in such a way that the value of  is as small as possible. Usually we choose for (2.40) the (global) minimal Weierstrass model (see [45, Section VIII.8]), which can be computed by various packages, for example, PARI, MAGMA, SAGE; not rarely, it happens that the minimal Weierstrass model is (4.1). The following proposition is useful in order to compute an upper bound for the canonical height for the point of an elliptic curve, if the coordinates of the point belong to a number field. Computing the canonical height of such a point is more difficult compared with the analogous task for a rational point (see [43]); anyway, standard packages such as PARI and MAGMA refuse to work with points over a number field. Fortunately, for the needs of this book it suffices that we know a “reasonably good” upper bound for the canonical height.

28

Chapter 2 Heights

Proposition 2.6.4. Let D : y12 C a1 x1 y1 C a3 y1 D x13 C a2 x12 C a4 x1 C a6 be a model of an elliptic curve, with a1 , a2 , a3 , a4 , a6 rational integers. Let P D 2 D.K/, where K is a number field. Then O /  h.x1 .P // C h.P

1 12

log jj C

1 12

logC jj j C 12 logC jb2 =12j C 12 log 2 ,

where, 1 and j1 are, respectively, the discriminant and j -invariant of the model D, b2 D a12 C 4a2 , 2 D 1 or 2 according to whether b2 vanishes or not, respectively, and logC is defined for any real ˛ > 0 by logC ˛ D log max¹1, ˛º. Proof. Immediate application of Silverman’s theorem [44, Theorem 1.1]. Note that, in our proposition, the coefficients of the equation defining D are rational integers.4 Therefore, 1 2 Z and j1 , b2 2 Q, so that we can replace h.1 / and h1 .j1 /, h1 .b2 =12/, appearing in Silverman’s theorem, by log j1 j and logC jj1 j, logC jb2 =12j, respectively.

4

Silverman’s theorem demands only that these coefficients be algebraic integers in K.

Chapter 3

Weierstrass equations over C and R

3.1

The Weierstrass } function

For the basic background of this section we refer to [1]. Let !1 , !2 be two complex numbers, such that !2 =!1 62 R, which we will call periods and let ƒ D Z!1 CZ!2  C be the lattice generated by these numbers, which we call a period lattice. The paral ! lelogram … D ¹x1 !1 C x2 !2 : 0  x1 , x2 < 1º and all its translations by vectors 0! 1 with ! 2 ƒ are called period parallelograms of ƒ. The lattice ƒ has infinitely many period parallelograms, since there are infinitely many bases .!1 , !2 / of ƒ; actually, any pair .!10 , !20 / obtained from .!1 , !2 / by a unimodular linear transformation with integer coefficients is another basis for ƒ. Definition 3.1.1. The Weierstrass }-function corresponding to the lattice ƒ is defined by the series X ..z  !/2  ! 2 /. (3.1) }.z/ D z 2 C !2ƒX¹0º

The Weierstrass }-function defined above is an even, doubly periodic function, with set of periods ƒ. It has a double pole at each ! 2 ƒ and is analytic at every z 2 C Xƒ; see [1, Theorem 1.10 ]. Since } is a meromorphic and doubly periodic function, it is, by definition, an elliptic function. Any basis .!1 , !2 / of ƒ is also called a fundamental pair of periods of }. If we translate the closure of a period parallelogram of ƒ by !  a vector 0z, (the end point z need not be a lattice point) and from each pair of parallel sides of the resulting (closed) parallelogram we remove exactly one side (and its vertices), we obtain a fundamental parallelogram of }. Thus, from the four vertices which bound a fundamental parallelogram, exactly one is contained in it. Obviously, any period parallelogram of ƒ is a fundamental parallelogram of the corresponding }-function but, clearly, the converse is not true. As noted just before Definition 3.1.1, for the function }, there are infinitely many fundamental pairs of periods, hence there are infinitely many period parallelograms and fundamental parallelograms. In view of periodicity it is clear that } may be viewed as an 1-1 function on any of its fundamental parallelograms rather than as a function on C.

1

Note that, in a period parallelogram, only the “left-lower” vertex is included; equally well, however, instead of this vertex, one could include a different one.

Chapter 3 Weierstrass equations over C and R

30

Let r D min¹j!j : ! 2 ƒ, ! ¤ 0º. Then, for 0 < jzj < r an alternative expression for }.z/ is 1 X 1 }.z/ D 2 C .2n C 1/G2nC2 z 2n , (3.2) z nD1 where, for n  3,

X

Gn D

!2ƒX¹0º

1 ; !n

(3.3)

see [1, Theorem 1.11]. The function } satisfies the differential equation } 0 .z/2 D 4}.z/3  g2 }.z/  g3 ,

(3.4)

where, by definition, g2 D 60G4 ,

g3 D 140G6 ;

(3.5)

see [1, Theorem 1.12]. Remark. }.z/, Gn , g2 and g3 are defined by means of a given lattice ƒ (of rank 2), therefore it would be more precise to respectively write }.z; ƒ/, Gn .ƒ/, g2 .ƒ/ and g3 .ƒ/. Actually, sometimes we will use this more precise notation if we want to emphasise the role of ƒ; otherwise, for the sake of simplicity in our notation, we will omit the indication of ƒ. Note that, by (3.1), }.z/0 D 2

X !2ƒ

1 , .z  !/3

which is an odd function. Further, the roots of the polynomial 4X 3  g2 X  g3 , which we will denote by e1 , e2 , e3 , are expressed in terms of the three non-zero half-periods, as follows: !1 !1 C !2 !2 / e3 D }. / (3.6) e1 D }. /, e2 D }. 2 2 2 and later on we will make use of the points def

Qi D .ei , 0/ 2 E.C/,

.i D 1, 2, 3/

(3.7)

The roots (3.6) are distinct, hence the discriminant  of the cubic polynomial 4X 3  g2 X  g3 is non-zero:  D g23  27g32 ¤ 0; see [1, §1.10]. By [1, Theorem 1.13], the coefficients G2nC2 in (3.2) are expressible as polynomials in g2 , g3 with positive rational coefficients, therefore the Weierstrass } function is uniquely defined by the pair .g2 , g3 / satisfying the condition g23  27g32 ¤ 0. The converse is also true. More precisely, if two given complex numbers a2 , a3 given satisfy a23  27a32 ¤ 0, then there exists a lattice ƒ D Z!1 C Z!2 , with !2 =!1 62 R, such that g2 .ƒ/ D a2 and g3 .ƒ/ D a3 ; see [1, Theorem 2.9].

31

Section 3.2 The Weierstrass equation

We summarise: Fact 3.1.2. Given a lattice ƒ D Z!1 C Z!2 , with !1 =!2 62 R, we define the Weierstrass function }.z; ƒ/ D }.z/ by means of (3.1), and the parameters g2 .ƒ/ D g2 def and g3 .ƒ/ D g3 by means of (3.5) and (3.3). Then, the discriminant .ƒ/ D  D g23  27g32 is non-zero and the differential equation (3.4) is satisfied. Conversely, given two complex numbers a2 , a3 such that a23  27a32 ¤ 0, there exists a lattice ƒ D Z!1 C Z!2 , with !1 =!2 62 R, such that g2 .ƒ/ D a2 and g3 .ƒ/ D a3 . The Weierstrass function }.z; ƒ/ D }.z/ defined by (3.1) will be called Weierstrass } function with parameters g2 D a2 and g3 D a3 . It is sometimes convenient to replace the lattice ƒ by its homothetic lattice ƒ D Z C Z , where D !2 =!1 , so that ƒ D !1 ƒ . The simple equations below relate the values of g2 , g3 and  that correspond to the lattices ƒ and ƒ . Note, however, that, instead of writing g2 .ƒ /, we simply write g2 . / and similarly for g3 and . We have g2 D g2 .ƒ/ D !14 g2 . / g3 D g3 .ƒ/ D !16 g3 . /

 D .ƒ/ D !112 . /

(3.8)

(see, for example, [1, §1.11]). Moreover, if is in the upper half-plane, then there is a Fourier expansion for . / 12

. / D .2/

1 X

t .n/e 2 i n ,

nD1

where t .n/ 2 Z for all n and a product expansion . / D .2/12 24 . /, where . / D e  i=12

1 Y

.1  e 2 i n /

(3.9)

(3.10)

nD1

(see [1, Theorem 3.3]).

3.2

The Weierstrass equation

Consider now an equation y 2 D x 3 C Ax C B,

A, B 2 C, 4A3 C 27B 2 ¤ 0

(3.11)

in complex numbers x, y. Since .4A/3  27.4B/2 ¤ 0, Fact 3.1.2 implies that there exists a lattice ƒ D Z!1 C Z!2 with g2 .ƒ/ D 4A and g3 .ƒ/ D 4B. The corresponding Weierstrass } function has parameters g2 D 4A and g3 D 4B.

32

Chapter 3 Weierstrass equations over C and R

Then, for every z 2 C X ƒ we have .} 0 .z/=2/2 D }.z/3 C A }.z/ C B, in view of (3.4). It is a very important fact that the converse is also true: If .x, y/ satisfies y 2 D x 3 C Ax C B (hence .2y/2 D 4x 3  g2 x  g3 ), then there exists a unique z mod ƒ such that .x, 2y/ D .}.z/, } 0 .z//. The proof is a consequence of a general property common to all elliptic functions, namely, that in a fundamental parallelogram with no poles or zeros on its boundary, the number of zeros is equal to the number of poles, with multiplicities of poles and zeros taken into account; see [1, Theorem 1.18]). Having this in mind we fix a pair .x, y/ as above and consider a fundamental parallelogram F. We will prove that there exists exactly one z 2 F such that .x, 2y/ D .}.z/, } 0 .z//. Obviously, it suffices to prove this property for any particular fundamental parallelogram. Therefore, for the needs of our proof we may assume that F is such that the points 0, !1 =2, !2 =2 and .!1 C !2 /=2 are contained in its interior. We consider first the case y D 0 which implies that x D ei for some i 2 ¹1, 2, 3º (cf. (3.6)). Depending on whether i D 1, 2 or 3, we see that, if w 2 F with w !1 !1 C!2 !2 , 2 , 2 .mod ƒ/, respectively, then the relation .ei , 0/ D .}.z/, } 0 .z// is sat2 isfied by z D w. Moreover, for fixed i 2 ¹1, 2, 3º, no other point z 2 F satisfies this relation. Indeed, since } is an elliptic function with 0 as its only pole in F, and this pole is double, it is not difficult to see that } 0 is an elliptic function with fundamental parallelogram F, its only pole in F is 0, and the order of this pole is 3. Hence, } 0 has 2 !2 , 2 are three distinct exactly three zeros in F. But we already know that !21 , !1 C! 2 0 zeros of } , hence these are the only ones; in particular, for the chosen i , there is no z 2 F other than w satisfying .ei , 0/ D .}.z/, } 0 .z//. Next, let y ¤ 0. Then we consider the function z 7! }.z/  x which, obviously, is elliptic and F, as above, is a fundamental parallelogram for this function. Since 0 2 F is a double pole for this function and no other poles exist in F, it follows that the equation }.z/  x D 0 has exactly two solutions z1 , z2 2 F (note that z1 , z2 62 2 !2 , 2 º). But since z1 is a solution, the same is true for the w 2 F satisfying ¹0, !21 , !1 C! 2 w z1 .mod ƒ/, since } is an even function. Thus x D }.z1 / D }.w/ and then, if in (3.4) we successively put z D z1 and z D w, we obtain 2y D ˙} 0 .z1 / and 2y D ˙} 0 .w/ with appropriate choice of signs. It follows that } 0 .w/ D ˙} 0 .z1 /, hence 2y is equal to either } 0 .z/ or to } 0 .w/. Note that w ¤ z1 .2 Therefore, necessarily, w D z2 , z2 z1 .mod ƒ/ and our discussion above showed that there exists exactly one z 2 F satisfying .x, 2y/ D .}.z/, } 0 .z// and this is either z D z1 or z D z2 . Our discussion is summarised as follows: Fact 3.2.1. Let A, B 2 C such that 4A3 C 27B 2 ¤ 0 and consider the elliptic curve model E : y 2 D x 3 CAxCB. Let } be the Weierstrass function with parameters g2 D 4A and g3 D 4B, let F be any fundamental parallelogram for } (in particular, a 2

Otherwise, 2z1 2 ƒ which easily implies that z1 2y D } 0 .z1 / D 0, a contradiction.

!1 !1 C!2 !2 , , 2 2 2

.mod ƒ/ and, consequently,

Section 3.3

: E.C/ 7! C=ƒ

33

period parallelogram) and let ! be the only pole of } in F. Then the map ´  .}.z/, 12 } 0 .z// if z ¤ ! F 3 z 7! 2 E.C/ O if z D !

(3.12)

is one-to-one and onto the group E.C/.3

3.3

: E.C/ 7! C=ƒ

In this section E will be the elliptic curve model considered in Fact 3.2.1. According to Section 1.1, E is just one out of many models of an elliptic curve E, therefore, any point on E, being a representative of a certain abstract point P (say) of E, should be denoted by P E . However, since in the present chapter we will not make use of any other model of E except for E, and for the sake of simplifying the notation, we will omit the superscript E on P , simply writing P instead of P E . Our main purpose is to define a group isomorphism the inverse of  that we defined in Fact 3.2.1.

: E.C/ 7! C=ƒ, which is

Let … be the period parallelogram with vertices 0, !1 , !2 , !1 C !2 and consider the map (3.12) with F D …. Obviously, we can view … as the abelian group C=ƒ. Then, Fact 3.2.1 is part of a far stronger result ([24, Theorem 6.17]) which says that the map  is a group isomorphism; even more, it is an isomorphism of Riemann surfaces (see [24, Theorem 6.14], or [45, Proposition 3.6]). We will not need the full strength of this result, but only that  is a group isomorphism. In view of Fact 3.2.1, for the proof of this very last claim it remains to prove that  is a homomorphism. It is more convenient to prove that the inverse map of , which we will denote by , is a homomorphism. First, let us make more explicit . As always, “zero point” is the point O. According to Fact 3.2.1, for any non-zero point .x, y/ 2 E.C/ there exists exactly one r 2 …, such that .x, y/ D .}.r/, 12 } 0 .r//. Therefore we have the well-defined bijection : E.C/ ! C=ƒ mapping O to ƒ and any non-zero point .x, y/ D .}.r/, 12 } 0 .r// to r C ƒ. We have to show that is a group homomorphism. We will make use of the socalled “addition formula” and “duplication formula” for Weierstrass } functions, referring, on the one hand to [73, §20.3] and, on the other hand, to the formulas for the addition and duplication of points on a Weierstrass model of an elliptic curve, as found, for example, in the “Group Law Algorithm 2.3” of [45]. Let P1 , P2 be two points on E.C/. Then, for i D 1, 2, there exists zi 2 … such that .x.Pi /, y.Pi // D .}.zi /, 12 } 0 .zi //. We have to prove that .P1 C P2 / D .P1 / C .P2 /. This is obviously true if at least one Pi is the zero point, therefore we assume that both P1 , P2 are non-zero points. We distinguish the cases }.z1 / ¤ }.z2 / and }.z1 / D }.z2 /. 3

Actually,  is a group isomorphism; see the beginning of next section.

Chapter 3 Weierstrass equations over C and R

34 (a) }.z1 / ¤ }.z2 /. Then,



 y.P2 /  y.P1 / 2  x.P1 /  x.P2 / x.P1 C P2 / D x.P2 /  x.P1 /   1 } 0 .r/  } 0 .z2 / 2  }.z1 /  }.z2 / D 4 }.z1 /  }.z2 / D }.z1 C z2 /,

the last equality being true because of the “addition-theorem” [73, §20.31]. On the other hand, by the first determinant formula in [73, §20.3], we have ˇ ˇ ˇ }.z1 / } 0 .z1 / 1 ˇˇ ˇ ˇ }.z2 / } 0 .z2 / 1 ˇˇ D 0 ˇ ˇ }.z1 C z2 / } 0 .z1 C z2 / 1 ˇ and after some standard calculations we find that 1 } 0 .z2 /  } 0 .z1 / 1 } 0 .z1 /}.z2 /  } 0 .z2 /}.z1 / 1 0 } .z1 C z2 / D  }.z1 C z2 /  2 2 }.z2 /  }.z1 / 2 }.z2 /  }.z1 / y.P1 /x.P2 /  y.P2 /x.P1 / y.P2 /  y.P1 / x.P1 C P2 /  D x.P2 /  x.P1 / x.P2 /  x.P1 / D y.P1 C P2 /. This proves that .x.P1 C P2 /, y.P1 C P2 // D .}.z1 C z2 /, 12 } 0 .z1 C z2 //, hence .P1 C P2 / D .z1 C z2 / C ƒ D .P1 / C .P2 /. (b) }.z1 / D }.z2 /. Then, } 0 .z2 / D ˙} 0 .z1 /. If the minus sign holds, then .}.z2 /, 12 } 0 .z2 // D .}.z1 /,  12 } 0 .z1 //, hence P2 D P1 and .P1 CP2 / D .O/ D ƒ. On the other hand, .P1 / C .P2 / D .z1 C z2 / C ƒ D ƒ and this last equality holds because .}.z2 /, 12 } 0 .z2 // D .}.z2 /,  12 } 0 .z2 // D .}.z1 /, 12 } 0 .z1 // which implies that z1 D z2 .mod ƒ/. If the plus sign holds, then z1 C ƒ D z2 C ƒ, hence, P1 D P2 D (say) P and we put .x.P /, y.P // D .}.r/, 12 } 0 .r// with r 2 P . We have then to show that .2P / D 2 .P /. If P D .ei , 0/ for some i 2 ¹1, 2, 3º (cf. (3.6)), then, on the 2 !2 , 2 º and, on the other hand, 2P D O. Consequently, one hand, z D z0 2 ¹ !21 , !1 C! 2 .2P / D ƒ D 2.z0 Cƒ/ D 2 .P /. If for every i D 1, 2, 3 we have P ¤ .ei , 0/, then y.P / ¤ 0 and } 0 .r/ ¤ 0. Then, differentiation of } 0 .r/2 D 4}.r/3  g2 }.r/  g3 D 4}.r/3 C 4A}.r/ C B gives } 00 .r/ D 2.3}.r/2 C A/. By the “duplication formula” [73, §20.311],     3}.r/2 C A 2 1 } 00 .r/ 2  2}.r/ /  2}.r/ D }.2r/ D 4 } 0 .r/ } 0 .r/ .3x.P /2 C A/2  8xy 2 D x.2P /. D 4y 2

Section 3.3

: E.C/ 7! C=ƒ

35

CA 2 We differentiate the relation }.2r/ D . 3}.r/ /  2}.r/ above and take into } 0 .r/ 00 2 account the relations } .r/ D 2.3}.r/ C A/, }.r/ D x.P / and } 0 .r/ D 2y.P /. After some elementary calculations we find that } 0 .2r/ D 2y.2P /. Thus, .x.2P /, y.2P // D .}.2r/, 12 } 0 .2r//, which shows that .2P / D 2r C ƒ D 2 .P /, as required. Combining our discussion above with Fact 3.2.1 we obtain the following theorem: 2

Theorem 3.3.1. We refer to Fact 3.2.1, in which we put F D …, where … is the period parallelogram at the beginning of this section. We identify F D … with the group C=ƒ. Then  : C=ƒ ! E.C/ is a group isomorphism, whose inverse isomorphism : E.C/ ! C=ƒ is as follows: .O/ D ƒ and, for a non-zero point P D .x, y/ 2 E.C/, we define .P / D r C ƒ, where r is the unique non-zero r 2 … with the property .}.r/, 12 } 0 .r// D .x, y/. Definition 3.3.2. Fix any fundamental parallelogram F of }. Theorem 3.3.1 implies that, for any P D .x, y/ 2 E.C/ there exists a unique r 2 F such that P D .}.r/, 12 } 0 .r//. This r is called elliptic logarithm of the point P (belonging to F). In analogy with complex logarithms, any point has infinitely many elliptic logarithms, depending on the fundamental parallelogram which we choose. Any two of them, however, differ by an element of ƒ. This situation is similar to that of complex logarithms, in which any non-zero complex number has infinitely many logarithms and any two of them differ by an integral multiple of 2 i . Now a natural problem arises: Problem 3.3.3. Given a point P D .x.P /, y.P // 2 E.C/ and a fundamental parallelogram F, compute the elliptic logarithm of P belonging to F. In other words, compute r 2 F such that .}.r/, 12 } 0 .r// D .x.P /, y.P //. Once again we turn to the classical book [73]. First, we cut the complex plane along a line segment joining any two of e1 , e2 , e3 and along any half-line with origin the remaining ei which (half-line) has no common points with the (closed) line segment. What remains then is a cut plane which we denote by C ; we understand that e1 , e2 , e3 are not points of C . If we fix some z0 2 C, such that 4z03  g2 z0  g3 ¤ 0, along with one of the two square roots of 4z03  g2 z0  g3 , let us denote it by w0 , then there exists an analytic function h on C such that h.z/2 D 4z 3  g2 z  g3 for every z 2 C and h.z0 / D w0 . This pis a very classical result. For obvious reasons, this analytic function h is denoted by 4z 3  g2 z  g3 , despite the ambiguity of the notation coming from the fact that, for the chosen z0 there are two choices for w0 . However, in using the square root notation, we understand that a certain pair .z0 , w0 /, as above, has been fixed. R1 , where the integration Suppose now that w 2 C . Let z.w/ D w p 3 dt 4t g2 tg3

path, except for 1 and, possibly, its end point w, is contained in C and does not pass

Chapter 3 Weierstrass equations over C and R

36

through any ei , i.e. only its end point w might coincide with a root ei . Then, according to [73, §20.221], }.z/ D w. This is the so-called integral formula for }.z/, which is expressed by the following implication: Z 1 dt p ) }.z/ D w. (3.13) zD 3 w 4t  g2 t  g3 Using (3.13) we can partially solve Problem 3.3.3. Indeed, setting in (3.13) w D x.P / we have }.z/ D x.P /. Consequently, } 0 .z/2 D 4x.P /3 g2 x.P /g3 D .2y.P //2 , implying "} 0 .z/ D 2y.P /, where " 2 ¹1, 1º. Then, since } 0 is an odd function, } 0 ."z/ D 2y.P /. If we choose r 2 F such that r D "z .mod ƒ/, then .x.P /, y.P // D .}.r/, 12 } 0 .r//, as required. This answer is partial in the sense that, given the point .x, y/ we do not know a priori how to choose " 2 ¹1, 1º. However, in the case that A and B are real numbers and P 2 E.R/ we will give a very satisfactory answer in Section 3.5.

3.4 Weierstrass equations with real coefficients In view of Fact 1.1.2, we intend to study Diophantine equations E : y 2 D x 3 C Ax C B,

A, B 2 Q,

4A3 C 27B 2 ¤ 0.

(3.14)

In terms of the Weierstrass function, we have to find all z 2 C, such that .}.z/, 12 } 0 .z// is an integral (or rational) point, where } is the Weierstrass function with parameters g2 D 4A,

g3 D 4B.

(3.15)

In such a case, .x, y/ D .}.z/, 12 } 0 .z// is a sought for point on E. It is natural, therefore, that we focus our study on Weierstrass }-functions whose period lattice ƒ D Z!1 C Z!2 is such that g2 D g2 .ƒ/ and g3 D g3 .ƒ/ are real numbers, forgetting for the moment that, in the context of our Diophantine study, these are actually rational numbers. In this section we will study the elliptic curve model E defined by the equation (3.11) when A, B 2 R. We will adopt the notation at the beginning of Section 3.2 and will refer to Fact 3.2.1, according to which every point .x, y/ 2 E.C/ is of the shape .x, y/ D .}.z/, 12 } 0 .z// for a unique z belonging to a fundamental parallelogram. The problem that will concern us in this section is the following: When A, B 2 R, to specify exactly those z in a conveniently chosen fundamental parallelogram for which the point .}.z/, 12 } 0 .z// has real coordinates, that is, it belongs to E.R/. In our study which we start immediately below, we will distinguish between the case of positive discriminant (Subsection 3.4.1) and the case of negative discriminant (Subsection 3.4.2). Finally, in Subsection 3.4.3 we will find explicit expressions for a fundamental pair of periods. To a large extent our exposition follows closely that of A. Hurwitz [23].

Section 3.4 Weierstrass equations with real coefficients

37

We have ƒ D Z!1 C Z!2 , where !1 =!1 62 R and both g2 D g2 .ƒ/ D 4A and g3 D g3 .ƒ/ D 4B are real. By definition, G4 D g2 =60, G6 D g3 =140 (cf. (3.5)). Moreover, by [1, Theorem 1.13], G2nC2 2 QŒg2 , g3 , hence, all coefficients of the series in the right-hand side of (3.2) are real, which implies that }.z/ D }.z/. We claim now that ƒ contains both real and purely imaginary periods. Surely, ƒ contains an ! such that 0. Case 2. Periods are included in the interior of the parallelogram P. Let ! D r C si 2 ƒ belong to the interior of P. Then, 0 < r < !1 and 0 < s < =.!2 /. According to our discussion above, ! 2 ƒ, hence 2r D ! C ! and 2s D !  ! are periods with 0 < 2r < 2!1 and 0 < 2s < 2=!2 . These conditions imply, respectively, 2 . It is easy to see then 2r D !1 and 2si D !2 ; hence ! is the centre of P: ! D !1 C! 2 that the parallelogram with vertices 0, !1 , !1 C !, ! contains no lattice points in its 2 is a fundamental interior, hence it is a period parallelogram of ƒ and !1 , ! D !1 C! 2 1 pair of periods with D !=!1 D 2 .1 C i t /, where i t D !2 =!1 2 i RC . Then, e 2 i D e  t 2 R and, consequently, by (3.9) and (3.10), . / is a negative real number. We summarise our conclusions: .!1 , !2 / is a fundamental pair of periods if and only if  > 0 and .1 , 2 / D 2 / is a fundamental pair of periods if and only if  < 0. .!1 , !1 C! 2 We examine the two cases ( > 0 and  < 0) separately.

Chapter 3 Weierstrass equations over C and R

38

3.4.1  > 0 In Figure 3.1 we translate the closure of the period parallelogram marked out by vertices , by the vector with initial point 12 .!1 C!2 / and end point 0, and then we remove the lower and the left side (with their vertices) of the resulting parallelogram. We thus obtain the fundamental parallelogram, the vertices of which are marked by ı in Figure 3.1; by its definition, this fundamental parallelogram contains only the upper-right vertex. We examine for which z 2 P we have }.z/ 2 R: }.z/ 2 R , }.z/ D }.z/ , }.z/ D }.z/ ) } 0 .z/ D ˙} 0 .z/. Therefore, if z 2 P and }.z/ 2 R, then the point .}.z/, } 0 .z// is equal to either .}.z/, } 0 .z// or .}.z/, } 0 .z//, hence, either z D z .mod ƒ/ or z D z .mod ƒ/; equivalently, 1 1 .z ˙ z/ 2 ƒ. (3.16) 2 2 Remember now that P includes only the right vertical and the upper horizontal side. If z is on one of these sides of P, then (3.16) is true with the plus or the minus sign, respectively. Next, consider points z in the interior of P. Since the only point of 12 ƒ lying in the interior of P is 0, it is easy to see that (3.16) is true with the plus or minus sign if and only if z is, respectively, on the line segment joining  !22 with !22 or the line segment joining  !21 with !21 . Now we consider the “small grey” closed parallelogram P0 with vertices 0, !21 , !1 C!2 !2 , 2 . Our discussion above, along with the fact that the function } is even, imply 2

ω2

ω1 + ω2

− 1 ω1 2 P

0

P0 1ω 2 1

fundamental parallelogram

− 1 (ω1 + ω2 ) 2

period parallelogram

1 (ω + ω ) 2 1 2

1 (ω − ω ) 2 1 2

ω1

1 (ω − ω ) 1 2 2

Figure 3.1. The fundamental parallelogram and the period parallelogram: Case  > 0.

Section 3.4 Weierstrass equations with real coefficients

39

that, as z runs along the contour of P0 , all possible real values of }.z/ are obtained. Therefore, we study the behaviour of the function } on every side of P0 . The function .0, !21  3 z 7! }.z/ 2 R is 1-1. Indeed, suppose that z1 ¤ z2 are on the domain of this function and }.z1 / D }.z2 /. Then } 0 .z2 / D ˙} 0 .z1 /. If the plus sign holds, then z1 D z2 .mod ƒ/, clearly impossible. If the minus sign holds, then .}.z2 /, } 0 .z2 // D .}.z1 /, } 0 .z1 // and, consequently, z2 C z1 2 ƒ, again impossible, since 0 < z1 Cz2 < !1 . Observe now that, in view of (3.2), limz!0C }.z/ D C1 and, in view of (3.6), }. !21 / D e1 . Therefore, the continuous function above is 1-1 and takes on all values in the interval .C1, e1 . It follows that e1 is the largest among the roots (3.6) of the polynomial 4X 3  g2 X  g3 . For, otherwise, some z 2 the interval .0, !21 / would exist such that }.z/ is one of the numbers (3.6), other than e1 . This 2 belongs to ƒ, which is impossible. would imply that either z  !22 or z  !1 C! 2 2 . Next consider the function } restricted to the side with end points !21 and !1 C! 2 With similar arguments we conclude the following: Along this side, the function } is 1-1; at the vertex !21 it takes the value e1 which, as already seen, is the least among 2 / (D e2 , by (3.6)) which, the roots (3.6); at the other vertex it takes the value }. !1 C! 2 therefore, must be the second larger root; consequently, e3 D }. !22 / is the least root of this polynomial. 2 and !22 the values of }.z/ Similarly, as z runs along the side with end points !1 C! 2 vary in 1-1 way from the middle root e2 to the smallest root e3 . Finally, as z runs along the side with end points !22 and 0, the values of }.z/ vary in a 1-1 way from the smallest root e3 to 1. Now we study the behaviour of the function } 0 . We have seen that }.z/ is strictly decreasing as z runs along the side with end points 0 and !21 , hence, for these z’s we have }.z/  e1 . Then, } 0 .z/2 is a non-negative real number, hence } 0 .z/ is a real number; more precisely, since }.z/ is strictly decreasing, we have } 0 .z/  0 with equality holding if z D !21 . Next, we see that the values of }.z/ as z runs along the points of the upper horizontal 2 to !22 , coincide with the values of the side of P0 following the direction from !1 C! 2 real function Œ0, !21  3 x 7! }.x C !12C!2 / 2 Œe3 , e2 , which is continuous, mapping 0 to e2 and !21 to e3 . Therefore, this is a strictly decreasing function, hence its derivative takes negative values in the open interval .0, !21 / and becomes zero at 0 and !21 . But the derivative values of the above function coincide with those of } 0 .z/ as z runs along the upper horizontal side of P0 , hence } 0 .z/ < 0 for z in the interior of this side and } 0 .z/ D 0 if z is a vertex. When z runs along the vertical sides of the parallelogram P0 , the values of }.z/ belong to either the interval Œe2 , e1  (if z is on the right vertical side) or to the interval .1, e3 / (if z is on the left vertical side), hence } 0 .z/2  0 and, except if z is an end point of a side, .}.z/, } 0 .z// 62 E.R/. Summing up: The only points z in P0 for which .}.z/, 12 } 0 .z// 2 E.R/ are those of the two horizontal sides. For those z’s, we have } 0 .z/  0, with equality only on

Chapter 3 Weierstrass equations over C and R

40

Table 3.1. Case  > 0. All points .}.z/, 12 } 0 .z// 2 E.R/. z runs along half-open line segments . !2 !1 2

...................

!2 C!1  2

. !1 . . . . . . . . . . . . . . . . . . . . . . . !1  2

2

z

!2 !1 2

Ý

!2 2

Ý

!2 C!1 2

 !1 2

Ý

0

Ý

!1 2

}.z/

e2

&

e3

%

e2

e1

%

C1

&

e1

} 0 .z/

0

>0

0

0, we see that }.z/ 2 R is equivalent to 12 .z ˙ z/ 2 P \ 12 ƒ. It can be easily checked that the last relation is true with the plus or the minus sign if and only if z is on the vertical or the horizontal diagonal of P, respectively. By (3.6), }. 21 / D }. !21 / D e1 , hence this is the only real root of 4X 3  g2 X  g3 . Now, assume that z runs along the horizontal diagonal of P. Then }.z/ 2 R and, by (3.2), limz!0 }.z/ D C1. Moreover, } is 1-1 on the intervals . 21 , 0/ and .0, 21 , respectively. Therefore, as z moves from  21 to 0 along the real axis, }.z/ increases from e1 to C1. In particular, this implies that } 0 .z/ > 0. On the other hand, as z runs along the interval .0, 21 /, }.z/ decreases from C1 to e1 which, in particular, implies that } 0 .z/ < 0. Next, assume that z runs along the vertical diagonal of P. If z tends to 2  12 1 , then }.z/ tends to }. 12 1 / D e1 . For the interior points z of the upper half of this diagonal we have, by (3.2), }.z/ < 0 and limz!0i }.z/ D 1, hence the values of }.z/ belong to the interval .1, e1 /. It follows that 4}.z/3  g2 }.z/  g3 < 0. Consequently, } 0 .z/ 2 i R and no real point on E arises when z is on the open upper half of the vertical diagonal of P. Similarly the same is true if z is in the open lower half of this diagonal. In Table 3.2 we schematically summarise our results.

3.4.3

Explicit expressions for the periods

We refer to our discussion, notation etc. following Problem 3.3.3 up to the end of Section 3.3. way, so that the real half-line from Computation of !1 . We cut C in the appropriate p e1 to C1 is contained in C and consider 4x 3  g2 x  g3 as the panalytic continuation on C of the positive real-valued function .e1 , C1/ 3 x 7! 4x 3  g2 x  g3 . With w D e1 in (3.13) and integration path the half-line from e1 to C1 we have

42

Chapter 3 Weierstrass equations over C and R

}.z/ D e1 D }. 12 !1 /, hence Z 1 dt !1 p 3 2 e1 4t  g2 t  g3

.mod ƒ/

(observe that  !21 !21 .mod ƒ/). Denote, temporarily, by I the integral in the right-hand side of the above relation. Since I 2 RC , this relation can be written as I D 12 !1 C n!1 for some non-negative integer n. We claim that n D 0. Suppose that n > 0. Consider the real function Z C1 dt p t 7! I.x/ D 3 x 4t  g2 t  g3 defined on the interval .e1 , C1/, which is continuous and strictly decreasing. Since limx!e1 C I.x/ D I , this function takes values in the interval .0, 12 !1 C n!1 /, therefore, there exists x0 > e1 such that I.x0 / D 12 !1 . This says that the relation (3.13) holds with w D x0 and z D 12 !1 , which implies that x0 D }.!1 =2/ D e1 , a contradiction. Therefore, n D 0 and Z C1 !1 dt p , (3.17) D 2 e1 4t 3  g2 t  g3 hence, in terms of the coefficients of our Diophantine equation (3.14), we obtain, in view of (3.15), Z C1 dt . (3.18) p !1 D e1 t 3 C At C B Computation of !2 . We distinguish the two cases  > 0 and  < 0. We put q.t / D 4t 3  g2 t  g3 . Let  > 0. In this case we choose C to be the complex plane from which we have removed the open line segment with end points e2 , e3 and the open half-line on axis are the intervals the real axis from e1 to C1. What remains in C from the real p  q.x/ is defined. .1, e3  and Œe2 , e1  on which the real non-negative function p We consider therefore the analytic function C 3 z 7! i  q.z/ 2 C which, for z 2 .1, e3  [ Œe2 , e1 , takes values in i RC . Clearly, this is an p analytic branch in of the square root of q.x/. Therefore, if in (3.13) we take i  q.t / in place of C p 3 4t  g2 t  g3 , e3 in place of w and, as an integration path, the half-line on the real axis from e3 to 1, then we have Z 1 dt p , }.z/ D e3 D }.!2 =2/. zD e3 i  q.t / It follows that Z 1 !2 dt i p .mod ƒ/. 2 e3  q.t /

43

Section 3.4 Weierstrass equations with real coefficients

Since !2 2 i R, the last relation is equivalent to Z e3 s dt p D C ns, where !2 D i s, s 2 RC and n 2 Z. 2 q.t / 1 The left-hand side is positive, therefore, n  0. We will show thatR n D 0. Indeed, x suppose n > 0 and consider the real continuous function I.x/ D 1 p dt , deq.t/

fined on the interval .1, e3 /. It is an increasing function with values in the interval .0, . 12 C n/s/. Therefore, there exists x0 < e3 such that I.x0 / D s=2, hence R 1 p dt D !2 =2 and, consequently, x0 D iI.x0 / D !2 =2. This means that x0 i

q.t/

}.!2 =2/ D e3 , a contradiction. Therefore, n D 0 and we conclude that: Z !2 D 2i

e3

1

Case  > 0 Z e3 dt dt p p Di . q.t / .t 3 C At C B/ 1

(3.19)

Let  < 0. Now we cut the complex plane along the half-line on the real axis from e1 to C1 and along the segment joining e2 with e3 D e2 . In analogy to the previous p case we observe that q.x/ > 0 for x < e1 , the real function .1, e1  3  q.x/ 2 Œ0, C1/ is well defined and is extended analytically on C . As x 7! R 1 p dt , then }.z/ D e1 D }.!1 =2/. Now, a fundamental pair of before, if z D e1 i q.t/ R e1 R 1 p dt p dt periods is .1 , 2 / D .!1 , .!1 C!1 /=2/, therefore i 1 D e1 D q.t/

i

q.t/

2 C m!1 C n !1 C! D . 12 C m C n2 /!1 C n2 !1 , for some m, n 2 Z. The left-hand 2 C side, as well R e1 as !dt2 belongn to i R , therefore, the coefficient of !1 must be zero, n is D s, where !2 D i s. In the last but one equality, the integral odd and 1 p

!1 2

q.t/

2

is positive, hence nR 1. If n  3, then, arguing as in the previous cases, we can find R 1 x0 p dt p dt x0 < e1 such that 1 D s2 , hence x0 D s2 and, consequently, q.t/ !1 / 2

2 x0 D }. !22 / D }. !1 C!  2 and we conclude that

i

q.t/

D }. !21 / D e1 , a contradiction. Therefore n D 1

Case  < 0 Z e1 dt Di . !2 D 2i p p 3 q.t / .t C At C B/ 1 1 Summing up we have the following theorem. Z

e1

dt

(3.20)

Theorem 3.4.1. Let q.X / D 4X 3  g2 X  g3 D 4.X 3 C AX C B/ D 4f .X /, where A, B 2 R and  D 16.4A3 C 27B 2 / ¤ 0. Denote by e1 , e2 , e3 the roots of q.X / – which are also the roots of f .X / – where e3 < e2 < e1 if  > 0, and e1 2 R, e3 D e2 if  < 0. Let } be the Weierstrass function with parameters g2 D 4A, g3 D 4B. Finally, let !1 be the least positive real period and let !2 be the totally complex period with least positive imaginary part. Then

Chapter 3 Weierstrass equations over C and R

44 

!1 is given by (3.17) or, equivalently, by (3.18);



!2 is given by (3.19) if  > 0 and by (3.20) if  < 0.

A fundamental pair of periods for } is .!1 , !2 / if  > 0 and .!1 , 12 .!1 C !2 // if  < 0.

3.4.4 Computing !1 and !2 in practice We keep the notation of Theorem 3.4.1. Our purpose in this section is to indicate a fast method for computing !1 and !2 . This will be accomplished by means of the arithmetic-geometric mean (AGM), about which we immediately state the basic facts. Let a, b be two positive real numbers. We define the sequences .an / and .bn / as follows:

anC1

a0 D a, an C bn D , 2

bnC1

b0 D b, p D an bn

.n  0/.

In view of bn  bnC1  anC1  an for n  1 we conclude that both sequences are convergent and then, taking limits in 2anC1 D an C bn we see that lim an D lim bn . This common limit is denoted by M.a, b/ and is called the arithmetic-geometric mean (AGM) of a, b. The computation of M.a, b/ is very fast in view of the relation anC1  bnC1 D 8b1 .an  bn /2 . The following formula is due to Lagrange and Gauss: 1

Z

2

p

0

ds a2 cos2 s C b 2 sin2 s

D

2 . M.a, b/

(3.21)

For an elementary proof we refer to [5]. We consider first the case  > 0. Our problem is the practical computation of the integrals in (3.17) and (3.19). In this case we follow [5, Section 2.1]. First we compute the integral in (3.17). The change of variable t0 D gives

Z

e2 t  e1 e2 C e1 e3  e2 e3 t  e2

C1 e1

dt p D q.t /

Z

e3

e2

dt p q.t /

and then, the change of variable t D e3 C .e2  e3 / sin2 s transforms the last integral into Z ds 1 2 , p 4 0 .e1  e3 / cos2 s C .e1  e2 / sin2 s p p which, by (3.21) is equal to 2=M. e1  e3 , e1  e2 /.

45

Section 3.4 Weierstrass equations with real coefficients

In a similar way we see that the integral in (3.19) is equal to p p turn, is equal to 2=M. e1  e3 , e2  e3 /. Therefore, by (3.17) and (3.19) we conclude:

R e1 e2

p dt

q.t/

which, in

Case  > 0. A fundamental pair of periods is .!1 , !2 /, where  p p M. e1  e3 , e1  e2 / i . !2 D p p M. e1  e3 , e2  e3 / !1 D

(3.22) (3.23)

Now, let  < 0. Our model E1 : y 2 D 4x 3  g2 x  g3 is 2-isogenous with the model E2 : Y 2 D 4X 3  4.15e12  g2 /X  2.7e1 g2 C 11g3 /,

(3.24)

via the isogeny  : E1 ! E2 defined by  3 4x  8e1 x 2 C .16e12  g2 /x  3g3  2e1 g2 .x, y/ 7! .X , Y / D , 4x 2  8e1 x C 4e12  8e12 y  g2 y  4yx 2 C 8e1 yx 4x 2  8e1 x C 4e12 with dual isogeny O : E2 ! E1 defined by  3 2X C 8e1 X 2 C 2.e12 C g2 /X  3g3 C e1 g2 .X , Y / 7! .x, y/ D , 8X 2 C 32e1 X C 32e12  g2 Y  7e12 Y  YX 2  4e1 YX . 8X 2 C 32e1 X C 32e12 The discriminant of E2 is equal to 2 D 18e12 g22 4g23 C27g32 27e1 g2 g3 D 4.12e12  g2 /.3e12  g2 /2 . The discriminant of E1 is 1 D  D .g23  27g22 /=16, hence 27g32  g22 > 0. Since g3 D 4e13 g2 e1 , the last inequality becomes .3e12 g2 /.12e12 g2 /2 > 0. Consequently, 3e12  g2 > 0 and a fortiori 12e12  g2 > 0. Thus 2 > 0. Moreover, we see that the three real roots of the polynomial in the right-hand side of (3.24) are p p (3.25) e1  12e12  g2 < 2e1 < e1 C 12e12  g2 . Since 2 > 0, we conclude that E2 has a fundamental pair of periods .w1 , w2 / with w1 2 RC and w2 2 i RC . Therefore we calculate w1 , w2 in a very efficient way using the AGM algorithm in analogy with (3.22) and (3.23), but now with the roots

Chapter 3 Weierstrass equations over C and R

46

p 2 (3.25) p in place of e3 < e2 < e1 , so that, in place p of e1  e3 we have 2p 12e1  g2 D 2 2 2 4 3e1 C A, in place of e1  e2 we have 3e1 C 2 3e1 C A, and p 3e1 C 12e1  g2 D p in place of e2  e3 we have 3e1 C 12e12  g2 D 3e1 C 2 3e12 C A. Therefore, w1 D

 q  q p 2M 4 3e12 C A , 12 3e1 C 2 3e12 C A

w2 D

i r

q 2M

4

3e12 C A ,

1 2

q  3e1 C 2 3e12 C A

(3.26)

!.

Now, how can we calculate a fundamental pair of periods for E1 ? Let ƒ1 , ƒ2 be the period lattices for E1 and E2 respectively. The fact that  : E1 ! E2 is a 2-isogeny implies that ƒ1 is a sublattice of ƒ2 D Zw1 C Zw2 of index 2. By Section 3.4.2 we know that ƒ1 D Z1 C Z2 with 1 2 RC and 1  12 1 2 i RC . Therefore     w1 1 DV , 2 w2 where V is a 2  2 matrix with integer entries and determinant 2. We easily check that every such matrix V is of the form V D UA, where U 2 SL2 .Z/ and A is one of the following matrices:       2 0 1 0 2 0 , , . 0 1 0 2 1 1 Therefore, without loss of generality, we may assume that .1 , 2 / has one of the following values: .w2 , 2w1 /,

.2w2 , w1 /,

.2w1 , w1 C w2 /.

The first two cases are easily excluded in view of the conditions 1 2 RC and 1  1  2 i RC , so that the third case remains. Instead of .1 , 2 / D .2w1 , w1 C w2 / 2 1 we prefer to take as a fundamental pair of periods .!10 , !20 / D .2 , 1  2 / D .w1 C w2 , w1  w2 /. Now we relate !1 , !2 with w1 , w2 . Since the pair .!10 , !20 / generates all periods, we have !1 D m!10 C n!20 D .m C n/w1 C .m  n/w2 for some integers m, n. Since !1 is a real period, we must have m D n and !1 D m.2w1 /. But 2w1 D !10 C !20 is a positive real period, while !1 is the least real period. Therefore, m D 1, !1 D 2w1 and, consequently, from (3.26) we obtain  !. (3.27) !1 D r q q 1 M 4 3e12 C A , 2 3e1 C 2 3e12 C A

Section 3.5

: E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1

47

With completely analogous arguments we conclude that !2 D M

 p 4

i . q p 3e12 C A , 12 3e1 C 2 3e12 C A

(3.28)

Since .!10 , !20 / D .w1 C w2 , w1  w2 / D ..!1 C !2 /=2, .!1  !2 /=2, we can take the last pair as a fundamental pair of periods. Note that .!10 C !20 , !10 / D .!1 , .!1 C !2 /=2 is also a fundamental pair of periods. Case  < 0. As a fundamental pair of periods we can take any of .

!1 C !2 !1  !2 , /, 2 2

.!1 ,

!1 C !2 /, 2

where !1 , !2 are given by (3.27) and (3.28), respectively.

3.5

: E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1

We keep the notation of Theorem 3.4.1 and go back to the end of Section 3.3. We will revisit Theorem 3.3.1 now focusing our interest exclusively on points of E.R/. Our first task will be, given a point P 2 E.R/, to compute explicitly .P /; see Theorem 3.5.1 below. Next, in Theorem 3.5.2 we will focus on the set E0 .R/ of real points .x, y/, such that x  e1 , showing that, actually, the restriction of to E0 .R/ establishes a group isomorphism between E0 .R/ and R=Z!1 . Finally, we will show that, in the case of a positive discriminant, this last isomorphism can be extended to a “two-to-one” group epimorphism E.R/ ! R=Z!1 . Note first that E0 .R/ is indeed a subgroup of E.R/. For, if  < 0, then E0 .R/ D E.R/. If  > 0, then E.R/ D E0 .R/ [ E1 .R/, where E1 .R/ is the bounded set of real points .x, y/, such that e3  x  e2 , i.e. the set of points on the “egg”, in pop terminology. It is geometrically obvious that any line of the plane having a common point with the “egg” must have “two” common points with it, where in “two” we also include double points, in which case the line is tangent to the egg. Consequently, if a line joins “two” points belonging to E0 .R/ (with the meaning of “two” as above), then the third point of intersection must also lie on E0 .R/ and not on the “egg”; for otherwise, the line would have four common points – multiplicities taken into account – with the cubic model, which is absurd. From this we conclude that E0 .R/ is closed under addition of points and, consequently, E0 .R/ is indeed a subgroup of E.R/, hence a subgroup of E.C/ as well. Moreover, the sum of two points on the “egg” is a point on E0 .R/ and the sum of a point on the “egg” and a point on E0 .R/ is a point on the “egg”; in other words the group E.R/=E0 .R/ is isomorphic to Z2 . First, let P D .x.P /, y.P // 2 E0 .R/. From Tables 3.1 and 3.2 and the conclusions of the relevant subsections we see that there exists a unique real number r in

Chapter 3 Weierstrass equations over C and R

48

the interval . 12 !1 , 12 !1  such that .x.P /, y.P // D .}.r/, 12 } 0 .r//. We find now an explicit formula for r. Consider the real integral Z Z C1 1 C1 dt dt . p D p zD 3 2 x.P / t 3 C At C B x.P / 4t  g2 t  g3 According to the discussion following the integral formula (3.13) for }.z/, with a convenient " 2 ¹1, 1º, we have .x.P /, y.P // D .}."z/, 12 } 0 ."z//. On the other hand, since P 2 E0 .R/, we have x.P /  e1 , therefore 0 < z  12 !1 , in view of (3.17). From Tables 3.1 and 3.2 we see that } 0 .z/  0, concluding thus that " D 1 or 1 according to whether y.P /  0 or y.P /  0, respectively. Therefore, in this case, Z C1 dt 1 , (3.29) r D  "P p 3 2 x.P / t C At C B where "P D 1 or 1, according to whether y.P / > 0 or y.P /  0, respectively. Next, let P D .x.P /, y.P // 2 E1 .R/. Note that this can occur only in the case of positive discriminant. In (3.7) we defined the points Qi D .ei , 0/ (i D 1, 2, 3). 2 and .Q3 / D !22 , Theorem 3.3.1 and (3.6) imply that .Q1 / D !21 , .Q2 / D !1 C! 2 therefore we may assume that P ¤ Q2 , Q3 . We refer to Section 3.4.1. From Table 3.1 and the conclusions of that section we see that there exists an r on the line segment 1 1 (excluded) and !2 C! (included), such that .x.P /, y.P // D with end points !2 ! 2 2 1 0 .}.r/, 2 } .r//. We will find this r. Observe first that P C Q2 2 E0 .R/ and y.P C Q2 / have the same sign as y.P /. We have, in view also of our conclusion about points of E0 .R/, ..P C Q2 / C Q2 / D .P C Q2 / C .Q2 / Z C1 !1 C !2 1 dt C C ƒ, D  "P p 2 2 x.P CQ2 / t 3 C At C B

.P / D

where "P D 1 or 1, according to whether y.P / > 0 or y.P /  0, respectively. Therefore we are looking for integers m, n such that ..P C Q2 / C Q2 / D .P C Q2 / C .Q2 / Z C1 !1 C !2 1 dt C p C m!1 C n!2 D  "P 3 2 2 x.P CQ2 / t C At C B

rD

1 1 belongs to the line segment with end points !2 ! and !2 C! as mentioned above. 2 2 !1 !1 !2 This condition is equivalent to  2 0 we also set Q2 D .e2 , 0/. Then the isomorphism of Theorem 3.3.1, restricted to the subgroup E.R/ of E.C/, is the monomorphism : E.R/ ! C=ƒ, given by E.R/ 3 P D .x.P /, y.P // 7! r C ƒ 2 C=ƒ, where r is as follows: if P 2 E0 .R/ then r is given by (3.29) and if P 2 E1 .R/ (which can happen only if  > 0) then r is given by (3.30), where in both cases, "P D 1 if y.P / > 0 and "P D 1 if y.P /  0. Finally, by Theorem 3.3.1, .x.P /, y.P // D .}.r/, 12 } 0 .r//. The isomorphism E.R/ $ C=ƒ that is guaranteed by Theorem 3.5.1 is certainly important. However, from this book’s practical point of view, it has the following disadvantage: If P 2 E1 .R/, then .P / is not a real number, as !2 makes its appearance. This would imply one more unknown integer involved in the linear form (4.4) that we will meet in Chapter 4.4 That linear form is the basic tool for Ellog and, as we will see in Chapter 9, where we will obtain an upper bound for M , this bound depends exponentially on the number of the integer unknowns involved in the linear form. A “slight modification” of can rid us of this disadvantage at the cost of losing the “oneto-one” property in the case  > 0 (only). More specifically, in this section we will l define a “two-to-one” epimorphism E.R/ $ R=Z!1 with which we will work in the following chapters; as it will turn out, actually, the linear form mentioned above is a linear form in l-values of points of E. Theorem 3.5.2. We keep the notation and assumptions of Theorem 3.5.1. In particular, for any P 2 E.R/ we consider the complex number r, such that P and r are related as described in the above theorem. We define the map l0 : E0 .R/ ! R=Z!1 by setting l0 .P / D r C Z!1 , 4

More precisely, instead of m0 !1 (see (4.4)) we would have m0 !1 C m00 !2 .

Chapter 3 Weierstrass equations over C and R

50

and the map l : E.R/ ! R=Z!1 as follows: ´ l0 .P / l.P / D l0 .P C Q2 /

if P 2 E0 .R/ if P 2 E1 .R/.

(3.31)

(a) The map l0 is a group isomorphism. (b) The map l is a group epimorphism; moreover, the relation l.P1 / D l.P2 / implies that either P2 D P1 or P2 D P1 C Q2 . Next, we identify R=Z!1 with the interval I D . !21 , !21 , so that the sum of two numbers in I is the usual sum of real numbers plus an appropriate integral multiple of !1 in order that the result falls in I. Then, for every P 2 E.R/ we will view l.P / as a number of I. (c) If K  R is a number field, P1 , : : : , Pk 2 E.K/ are Z-linearly independent points of infinite order and T 2 E.K/ is torsion point, then, a relation n1 l.P1 / C    C nk l.Pk / C l.T / C n0 !1 D 0, where n0 , n1 , : : : , nk 2 Z is possible only if n1 D : : : D nk D 0. Proof. Two preliminary observations: First, if P 2 E.R/, then l0 .P / C ƒ D .P /. Indeed, let .P / D r C ƒ. By the definition of l0 , we have l0 .P / D r C Z!1 , therefore, l0 .P / C ƒ D r C Z!1 C ƒ D r C ƒ D .P /. Second, if P , P 0 2 E.R/ and l0 .P / D l0 .P 0 /, then P D P 0 . Indeed, in view of the previous observation, we have .P / D l0 .P / D l0 .P 0 / D .P 0 / and, since is one-to-one, it follows that P D P 0 . Proof of (a). Using the first preliminary observation and the fact that is a homomorphism, we show that l0 is a group homomorphism, as follows. Let P1 , P2 2 E0 .R/. Then, l0 .P1 C P2 / C ƒ D

.P1 C P2 / C ƒ D

.P1 / C

.P2 / C ƒ

D . .P1 / C ƒ/ C . .P2 / C ƒ/ D .l0 .P1 / C ƒ/ C .l0 .P2 / C ƒ/. Let .Pi / D ri C ƒ (i D 1, 2) and .P1 C P2 / D r C ƒ, where r1 , r2 and r are defined according to Theorem 3.5.1; by that theorem, r1 , r2 , r 2 R. On the other hand, l0 .Pi / D ri C Z!1 (i D 1, 2) and l0 .P1 C P2 / D r C Z!1 , therefore the above displayed relation implies that r  r1  r2 2 ƒ D Z!1 C Z!20 , where !20 D !2 if  > 0 and !20 D .!1 C !2 /=2 if  < 0. Let us write r  r1  r2 D m!1 C n!20 , where m, n 2 Z. The fact that the left-hand side is real and !2 2 i R forces n D 0 and r  r1  r2 D m!1 2 Z!1 , which is equivalent to l0 .P1 C P2 / D l0 .P1 / C l0 .P2 /. We show that l0 is one-to-one. If P 2 E0 .R/ and l0 .P / D 0 C !1 Z, then, by the first preliminary observation, .P / D 0 C ƒ. Since is one-to-one, it follows that P D O.

Section 3.5

: E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1

51

Finally, is also onto, for, if r is a real number and we set P D .}.r/, 12 } 0 .r// (if r 2 ƒ we mean, of course, that P D O) then, by Theorem 3.5.1, .P / D r C ƒ and, consequently, by the definition of l0 , we have l0 .P / D r C Z!1 . Proof of (b). The map l is a group epimorphism because, on the one hand, its restriction to E0 .R/ is, in view of (a), a group isomorphism and, on the other hand, the group E.R/=E0 .R/ is isomorphic to Z2 . For example, if P1 , P2 2 E1 .R/ then P1 C P2 2 E0 .R/ and, therefore, l.P1 C P2 / D l0 .P1 C P2 /. On the other hand, l.P1 /Cl.P2 / D l0 .P1 CQ2 /Cl0 .P2 CQ2 / D l0 .P1 CP2 C2Q2 / D l0 .P1 CP2 CO/ D l.P1 C P2 /, as required. Finally, let P1 , P2 2 E.R/ with l.P1 / D l.P2 /. If P1 , P2 2 E0 .R/, then l0 .P1 / D l.P1 / D l.P2 / D l0 .P2 /, therefore, by the second preliminary observation, P1 D P2 . Next, suppose that  > 0. If P1 , P2 2 E1 .R/, then, Pi C Q2 2 E0 .R/ for i D 1, 2, so that l0 .P1 CQ2 / D l.P1 / D l.P2 / D l0 .P2 CQ2 / and, as before, P1 CQ2 D P2 CQ2 , from which P1 D P2 . If exactly one among P1 , P2 , say the second, belongs to E1 .R/, then l0 .P1 / D l.P1 / D l.P2 / D l0 .P2 C Q2 /, hence P1 D P2 C Q2 , which is equivalent to P2 D P1 C Q2 . Proof of (c). Let n1 l.P1 / C    C nk l.Pk / C l.T / C n0 !1 D 0. Since l is a homomorphism, we have l.n1 P1 C    C nk Pk C T / D n1 l.P1 / C    C nk l.Pk / C l.T / C `!1 , for an appropriate integer ` which makes the left-hand side falling in the interval I D . 12 !1 , 12 !1 . In view of the hypothesis, the right-hand side of the last equality is equal to .`  n0 /!1 ; since the left-hand side belongs to I , this forces ` D n0 and l.n1 P1 C    C nk Pk C T / D 0. Then, in view of (b), n1 P1 C    C nk Pk C T is equal either to O or to Q2 . The second alternative is impossible if Q2 62 E.K/; and if Q2 2 E.K/, both alternatives imply that n1 P1 C  Cnk Pk is a torsion point of E.K/, which contradicts the hypothesis that P1 , : : : , Pk are Z-linearly independent.

Conclusions and remarks (1) In all subsequent applications of this book we will identify R=Z!1 with the interval I D . !21 , !21 , as explained in the announcement of Theorem 3.5.2. Note that, with this convention, l.P / is identified with the elliptic logarithm of P or of P C Q2 , depending on whether P belongs to E0 .R/ or to E1 .R/, respectively. Moreover, the discussion of this section which led us to (3.31) along with Theorem 3.5.1 immediately imply the following expression of l.P / as an integral: ´ Z "P C1 dt x.P / if P 2 E0 .R/ (3.32) , xP D l.P / D  p 2 xP x.P C Q2 / if P 2 E1 .R/ f .t / where, as before, "P D 1 if y.P / > 0 and "P D 1 if y.P /  0 (note that y.P / and y.P C Q2 / have equal signs).

Chapter 3 Weierstrass equations over C and R

52

(2) Let f .X / have rational coefficients. Since Q2 D .e2 , 0/ and e2 is a root of f .X /, we see that Q2 has coordinates in an extension of Q of degree at most three. Therefore, in view also of (1), above, we conclude that: If P 2 E0 .Q/, then l.P / is equal to the elliptic logarithm of a point (this is P ) with rational coordinates. If P 2 E1 .Q/, then l.P / is equal to the elliptic logarithm of a point (this is P CQ2 ) with coordinates belonging to a number field of degree at most three. (3) We defined the elliptic logarithm by means of a short Weierstrass model. In the chapters that follow, we often work with two different models of the same elliptic curve E, namely, a short Weierstrass model E and another model, say C . Therefore, in accordance with Section 1.1, any abstract point P of E has a representative P E on E and a representative P C on C , the coordinates of which satisfy the equations of E and C , respectively. Having fixed the short Weierstrass model E, we will always understand as elliptic logarithm of a certain point P of E the elliptic logarithm of P E . Therefore we will always omit the superscript E from the arguments of l and will write l.P /, instead of l.P E /. Practical computation of l.P/. Obviously, it suffices to compute l0 .P / for any point P 2 E0 .R/. We present here an algorithm due to D. Zagier [74]. Let l.P / D r.P / C Z!1 , where r.P / D r is given by (3.29) if P ¤ O and r.P / D 0 if P D O. In view of (3.18) and (3.29), r.P / 2 . !21 , !21 . We set now 8 ˆ 0 if P D O ˆ < r.P / if P ¤ 0 and y.P /  0 . s.P / D !1 ˆ ˆ : r.P / C 1 if P ¤ 0 and y.P / > 0 !1

Clearly, s.P / 2 .0,

1 / 2

if y.P / < 0; s.P / D

1 2

if y.P / D 0 and s.P / 2 . 12 , 1/ if

y.P / > 0. Further, s.P / D l0!.P1 / .mod Z/. Since l0 is a homomorphism, if P1 , P2 2 E0 .R/, then s.P1 CP2 / D s.P1 /Cs.P2 /Ck, where k 2 ¹1, 0, 1º and, consequently, for P 2 E0 .R/ and n 2 Z, s.n  P / D n  s.P / C an with an 2 Z. In order to compute l.P / it suffices to compute s.P /. If P ¤ O we write s.P / as a binary number s.P / D d21 C d222 C d233 C    , where di 2 ¹0, 1º for all i ’s. The digits d1 , d2 , d3 , : : : are computed successively as follows. If y.P / D 0 then d1 D 1, di D 0 for every i > 1 and the calculation of s.P / is complete. If y.P / < 0 then 0 < s.P / < 12 , forcing d1 D 0; and if y.P / > 0 then s.P / > 12 , hence d1 D 1. Suppose that, for some i  2 we have already computed d1 , : : : , di1 . We have d s.2i1  P / D 2i1 s.P / C k with k 2 Z, hence s.2i1  P / D ai1 C d2i C i2C1 C 2    , where ai1 2 Z. If 2i1  P D O, then s.2i1  P / D 0 and this can happen only if ai1 D 1 and dj D 1 for all j  i . Now suppose that 2i1  P ¤ O. If y.2i1  P / D 0 then s.2i1  P / D 12 , implying ai1 D 0, di D 1, dj D 0 for every j > i and the calculation of s.P / is finished. If y.2i1  P / < 0 then

Section 3.5

0 < ai1 C

: E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1 di 2

C

di C1 22

C 
0 then < ai1 C d2i C i2C1 C    < 1 which can occur only if 2 ai1 D 0 and di D 1. The above discussion suggests the following algorithm:

y.2i1

1 2

13 7 Computing the elliptic logarithm DESCRIPTION: Compute the elliptic logarithm l.P / D l0 .P / of P 2 INPUT: P 2 E0 .R/, N D number of desired binary digits for l.P /. OUTPUT: l.P /. INITIAL VALUES: u 0, s 0, Q P, i 1.

while i  N do if Q D O then goto () else if y.Q/ D 0 then s s C 1i : goto () 2 else if y.Q/ > 0 then s s C 1i end if 2 end if end if if i < N then Q D 2  Q endif i i C1 end while () u D !1 s. if P ¤ O and y.P / > 0 then u u  !1 end if l.P / D u. END

E0 .R/.

Chapter 4

The elliptic logarithm method

In this chapter we make a general description of the elliptic logarithm method, briefly Ellog, which will be used in later chapters for solving various types of elliptic equations in integers. In this chapter we will refer to the resolution of any such equation as the Diophantine problem. To each Diophantine problem, we will attach a convenient short Weierstrass model E : y 2 D x 3 C Ax C B,

A, B 2 Q;

(4.1)

this is done separately for each problem in Chapters 5, 6, 7, 8 and 11, respectively. We denote by r the rank of the elliptic curve E.Q/. We also denote by r0 the least common multiple of the orders of the non-zero points of Etors .Q/; by Mazur’s theorem (see [29, 30], or [45, Theorem 7.5]), 1  r0  12, r0 ¤ 11. We refer to Theorem 1.2.1, which we apply for K D Q. According to that theorem, if r > 0, then there exist points of infinite order P1E , : : : , PrE in E.Q/, with the property that every point P E D .x.P /, y.P // 2 E.Q/ is written uniquely in the form (4.2) P E D m1 P1E C    C mr PrE C T E for some integers m1 , : : : , mr and T E D .x.T /, y.T // 2 Etors .Q/. 11 Thus, (4.2) establishes a map P E ! .m, T E /, where m D .m1 , : : : , mr / is a lattice point in Rr and T E 2 Etors .Q/. A first important feature of Ellog is that the solutions of the Diophantine problems in Chapters 5, 6, 7, 8 and 11 are associated in some way, particular to each Diophantine problem, to points P E $ .m, T E / with m belonging to a (finite) cube of Rr ; that is, if P E is the point that “covers” (comes from) a solution of the Diophantine problem def and P E is expressed as in (4.2), then M D max1ir jmi j is bounded. Our task will be to compute an explicit upper bound for M ; a huge one first, and then with the use of it, a bound of “manageable” size. A second important feature of Ellog is that, given any point P E 2 E.Q/, one can easily (in principle) check whether it “covers” an actual solution of the Diophantine problem. Therefore, if an explicit upper bound, say K, of M is known, then, for each choice K  mi  K, i D 1, : : : , r and T E 2 Etors .Q/, a point P E 2 E.Q/ is calculated by means of (4.2) and then, this P E is checked whether it comes from a solution of the Diophantine problem. In particular, one can explicitly find the set T consisting of all points T of E, such that T E 2 Etors .Q/ and T covers a solution of

Chapter 4 The elliptic logarithm method

55

the Diophantine problem. As the calculation of the set T is a rather trivial task, we will henceforth assume that, once we are given a certain Diophantine problem, T has been computed already from the beginning and we are left with the difficult task of explicitly computing a finite set P of points of E with the following properties: Each solution of the Diophantine problem not covered by points of T , is covered by some point of P, and P E 62 Etors .Q/ for every P 2 P. The last condition is, of course, equivalent to (4.3) r  1 & .m1 , : : : , mr / ¤ .0, : : : , 0/. In this way we will effectively solve the Diophantine problem under consideration. Here, “effectively” means “after finitely many steps”, no matter how many they are. Take a very modest typical example, in which r D 2, the torsion points are two and, by some means we calculate K D 1030 , which is rather “very small”. Then the number of checks will be, approximately, 23  1060 . Of course, there is no hope to solve explicitly our Diophantine problem this way, but, at least, we have an effective or, if you prefer, a constructive proof of the fact that there exist finitely many solutions. Certainly, this is far more satisfactory for our mind than an existential (in other words, non-constructive) confirmation which would merely assure the existence of finitely many integer solutions. At the same time, an effective proof of the finiteness of solutions suggests, at least in theory, a method of resolution of the problem. We will show that one can exploit this effective method to finally obtain an explicit method of resolution of the Diophantine problem, i.e. a method whose output will be a finite set S consisting precisely of all solutions of the Diophantine problem. Below we make a very general description of how an upper bound for M is obtained. We recommend the reader to revisit very quickly Chapter 1 in order to recall the terminology, conventions and facts that will be used. In accordance with them: We will view E as a model of an elliptic curve. Referring to (4.1), we put g2 D 4A, g3 D 4B and we adopt the notations and assumptions of Theorem 3.4.1. In particular, q.X / D 4X 3  g2 X  g3 and e1 , e2 , e3 denote the roots of q.X /, where e1 > e2 > e3 if all three roots are real; otherwise e1 is the only real root. Further, we put f .X / D q.X /=4 D X 3 C AX C B. In order that the group relation (4.2) provide us with a numerical relation, we apply the group epimorphism l : E.R/ ! R=Z!1 of Theorem 3.5.2. We remind here that, according to this theorem, we view the group R=Z!1 as the interval . !21 , !21  in which the group operation is the usual addition “adjusted” by an integral multiple of !1 , so that the final sum falls in the above interval. The group relation (4.2) implies the following numerical relation1 0 ¤ l.P / D m1 l.P1 / C    C mr l.Pr / C l.T / C m0 !1 , 1

(4.4)

Below we omit the indication E from the points when they are an argument of the elliptic logarithm function l; see Conclusions and remarks 3, page 52.

56

Chapter 4 The elliptic logarithm method

where m0 !1 is the “adjusting summand” mentioned a few lines above. This is a linear form in elliptic logarithms of points with algebraic coordinates of degree at most three; see (2) in Conclusions and remarks, page 52. The non-vanishing of l.P / is justified by (4.3) and Theorem 3.5.2 (c). If t  2 is the (group) order of a torsion point T E ¤ O, then t  l.T / D l.t  T / D l.O/ D 0 2 R=Z!1 , which shows that l.T / D st !1 with  12 < st  12 . Thus, in general, r0 1 s 1 s (4.5) l.T / D , t jr0 , t > 0, jsj  ,  <  . t 2 2 t 2 Therefore, we can rewrite (4.4) as  s (4.6) 0 ¤ l.P / D m0 C !1 C m1 l.P1 / C    C mr l.Pr /. t Since every l-value is absolutely less than !21 , it follows from (4.6) and (4.5) that jm0 j  1 C .jm1 j C    C jmr j/=2. Therefore, if we set M D max jmi j, 1ir

then

M0 D max jmi j 0ir

® ¯ M0  max M , 12 rM C 1 .

(4.7)

(4.8)

Closely related to l.P / is a linear form L.P /, which we will define for each particular case in Chapters 5, 6, 7, 8 and 11. In each case we will compute an explicit upper bound for jL.P /j in terms only of M ; denote it (temporarily) by Ub.M /. On the other hand, the application of Theorem 9.1.2 to L.P / will furnish either an explicit upper bound for M itself, and we are done, or an explicit lower bound for jL.P /j in terms only of M , which we will temporarily denote by Lb.M /. In the second case we will thus arrive to Lb.M /  jL.P /j  Ub.M /. Both functions Lb.x/ and Ub.x/ are decreasing for x positive and sufficiently large. What is very important is that there exists an explicit positive constant K such that Lb.x/ > Ub.x/ for every x > K. This, combined with the last displayed relation, clearly implies that M  K, and we obtain the required upper bound for M . Remark on terminology. In the chapters that follow, the term “constant” always means “number depending on the particular Diophantine problem”. Once the Diophantine problem is expressed in concrete numerical terms, the “constant” has a specific numerical value. In the constants of the form ci , which are introduced in this book, we use subscripts i  7 in order to avoid confusion with the standard symbols c4 and c6 , that are used in the general theory of elliptic curves (see [45, p. 42]).

Chapter 5

Linear form for the Weierstrass equation

In this chapter we study the Weierstrass model of an elliptic curve C : g.u, v/ D 0 ;

g.X , Y / D Y 2 Ca1 X Y Ca3 Y .X 3 Ca2 X 2 Ca4 X Ca6 /, (5.1)

where a1 , a2 , a3 , a4 , a6 are rational integers. According to our discussion in Chapter 4, to the generic point P C D .u.P /, v.P // with integer coordinates we will attach a linear form in elliptic logarithms L.P / (cf. page 56 immediately after (4.8)). Our purpose is to prove Theorem 5.2, which gives an explicit upper bound for jL.P /j. This will be the first main step towards the explicit determination of all points P C as above. The presentation in this chapter is based on joint work of R. Stroeker and the author [54]. Throughout the chapter we keep the notations, assumptions and results of Chapter 4. For the basic background of this chapter we refer, explicitly or not, to [45, Chapter III]. As mentioned in Section 1.2, since C is a model of an elliptic curve, its discriminant C is non-zero. In the special important cases a1 D a2 D a3 D 0 and a4 D A, a6 D B, the discriminant is C D 16.4A3 C 27B 2 / ¤ 0 and we have seen in Section 3.1 how crucial the non-vanishing of this discriminant is. A linear substitution u D 2 x C ,

v D 3 y C 2 x C ,

(5.2)

with conveniently chosen rationals ¤ 0, , , , makes the model (5.1) birationally equivalent to a short Weierstrass model E defined by (4.1). The two discriminants,  of the model C , and E D 16.4A3 C 27B 2 / of the model E are related by  D 16 12 .4A3 C 27B 2 /. For example, a possible choice for , , ,  is . , , , / D 1 2 1 3 a1  13 a2 ,  12 a1 , 24 a1 C 16 a1 a2  12 a3 /. Thus, C and E are models of the .1,  12 same elliptic curve, say E. A basic fact is that, given a point P 2 E with u.P / 2 Z, sufficiently large, then its O / is “close” to the half of the logarithm of x.P /. This is expressed canonical height h.P by the following proposition: Proposition 5.1. Let ,  be as in (5.2) and let ı be the least common multiple of the denominators of 2 and = 2 . Then, for any P 2 E with u.P / an integer  2 C , O /  1 log x.P /   C 1 log ı, h.P 2 2 where  is defined by (2.41).

58

Chapter 5 Linear form for the Weierstrass equation

Remark. The proposition requires that the integer u.P / is at least 2 C . This is not a serious restriction in practice, because, as is easily seen, the points P with u.P / an integer < 2 C  are finitely many and can be explicitly computed very easily. Proof. We have x.P / D .u.P //= 2  1, hence, by the definition of ı, ı x.P / D .ı= 2 /  u.P /  ı.= 2 / is an integer. Therefore, if we write x.P / D a=b, where a, b are relatively prime positive integers, then bjı and, in particular, b  ı. Now, applying (2.21), we have h.x.P // D log max¹a, bº D log a, because x.P /  1. But 0 < a  ı  .a=b/ D ı  x.P /, therefore, h.x.P //  log ı C log x.P /. Now we apply Proposition 2.6.3 with C in place of D and .u, v/ in place of .x1 , y1 /. By (2.42) and the above relation we have O /  1 log x.P /  1 log ı, O /  1 h.x.P //  h.P   h.P 2

2

2

as claimed. The main result of this section is the following. Theorem 5.2. Let E be the elliptic curve represented by the model C in (5.1). Let E : y 2 D f .x/ be the short Weierstrass model (4.1) which is obtained from C by means of the transformation (5.2), so that everything in Chapter 4, especially relations (4.4) through (4.8), refers to this particular model E. Assume that P 2 E is such that P C D .u.P /, v.P // has integer coordinates with u.P / > 2 2 max¹ 12 , je1 j, je2 j, je3 jº C , where, as usual, e1 , e2 , e3 are the roots of f .X /. Write P E as in (4.2). With l.P / as in (4.6), set L.P / D l.P /. Then, with M , ,  and ı defined, respectively, by the first relation (4.7) and by Propositions 2.6.2, 2.6.3 and 5.1, we have p (5.3) jL.P /j  2 2  exp. C 12 log ı  M 2 /. Proof. First we observe that all x  2  max¹je1 j, je2 j, je3 jº satisfy the inequality Z 1 p dt 0<  4 2jxj1=2 . (5.4) p x f .t / Indeed, for such an x and t  x we have 0 < f .t / D jt  e1 jjt  e2 jjt  e3 j. As t is larger than the absolutely largest root of the polynomial f .X /,p it follows that jt e1 j  t  je1 j  t =2, and likewise for e2 and e3 . Consequently, 1= f .t /  23=2 t 3=2 and hence, for all N > x, Z N Z N p dt  p 23=2 t 3=2 dt D 4 2.x 1=2  N 1=2 /. 0< x x f .t / Letting N tend to infinity we obtain (5.4).

Chapter 5 Linear form for the Weierstrass equation

59

Now, in view of (5.2) and the hypothesis about the lower bound of u.P / we have x.P / > 2  max¹je1 j, je2 j, je3 jº, therefore, by (5.4), Z 1 p dt 0<  4 2  x.P /1=2 . p x.P / f .t / But, x.P /  e1 , therefore P E 2 E0 .R/ and consequently, by the definition of l (see (3.32)), thepleft-hand side in the above displayed relation is equal to 2jl.P /j, so that jl.P /j  2 2  x.P /1=2 . By Propositions 5.1 and 2.6.2, 

log¹x.P /

1 2

O /   C 1 log ı  M 2 , º   C 12 log ı  h.P 2

which completes the proof. To continue with the resolution of the Weierstrass equation, one can safely avoid Chapters 6, 7 and 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.3.

Chapter 6

Linear form for the quartic equation

In this chapter we will deal with the integer solutions of Diophantine equations F .U / D V 2 , where F denotes a quartic polynomial with rational coefficients and the equation F .x/ D y 2 defines an elliptic curve over Q. This implies, first, that the genus of the corresponding curve is one, which is equivalent to the non-singularity of the curve, hence to the non-vanishing of the discriminant of the polynomial F .X /, and second, that we know a rational solution .U , V / D .u0 , v0 /. In practice, the search for such a solution can be done by some powerful computational tool like ratpoints [53] due to M. Stoll [52], or the routine Points of MAGMA; of course, although it is highly probable that the search will be successful, there is no guarantee for this. Anyway, assuming this, we then reduce our equation to solving in integers the equation Q.u/ D v 2 , where Q.u/ D F .u C u0 /, .u, v/ D .U  u0 , V / and the constant term of the polynomial Q is a square of an integer. Therefore, we will consider the elliptic curve E, a model of which is on the quartic elliptic model

C : g.u, v/ D 0,

g.u, v/ D v 2  Q.u/,

Q.u/ D au4 C bu3 C cu2 C du C e 2 ,

(6.1)

a, b, c, d , e 2 Q , a, e > 0 , discr.Q/ ¤ 0. (6.2)

Before proceeding we note that we may assume that a is not a perfect square, for, in this case, the equation Q.u/ D v 2 in integers can be solved very efficiently by totally elementary means; see [37]. According to our discussion in Chapter 4, to P C D .u.P /, v.P //, the generic point on C with integer coordinates, we will attach a linear form in elliptic logarithms L.P /; see page 56 immediately after (4.8). We intend to prove Theorem 6.8, which gives an explicit upper bound for jL.P /j, as a first main step towards the explicit determination of all points P C as above. Our presentation is based on the author’s paper [62]. Throughout the chapterpwe keep the notations, assumptions and results of Chapter  is used, always, care is taken that the radicand be non4. Wherever the symbol p negative and then  denotes the non-negative square root of the radicand. The short Weierstrass model (4.1) with 1 A D  c 2 C bd  4ae 2 , 3

BD

2 3 1 8 c  bcd  ace 2 C b 2 e 2 C ad 2 27 3 3

(6.3)

61

Chapter 6 Linear form for the quartic equation

is birationally equivalent to (6.2) and the birational transformations (1.3) and (1.4) between the two models (cf. Fact 1.1.2) are the following: 6ev C cu2 C 3du C 6e 2 , 3u2 ebu3 C 2ecu2 C 3edu C duv C 4e 2 v C 4e 3 y D Y.u, v/ D  u3

(6.4)

u D U.x, y/ D .12e 2 x C 8ce 2  3d 2 /=.6ye  3dx C cd  6be 2 /

(6.5)

x D X .u, v/ D

1 v D V.x, y/ D .32c 3 e 3  108b 2 e 5 C 108bcde 3 C 27 d 3 y C 216 be 4 y 3  108 de 2 cy C 216e 3 x 2 c  81ex 2 d 2 C 9ec 2 d 2 C 216e 3 x 3  27bed 3  108e 3 y 2 /=.6ye  3dx C cd  6be 2 /2 .

(6.6)

In the above relations,  D ˙1; for both  D C1 and  D 1 a valid birational transformation is obtained. However, for the needs of this chapter it will be useful to chose the value of  according to the following lemma. Lemma 6.1. At least one of the two numbers p p d a C eb, 8e 3 a C 4e 2 c  d 2 is non-zero, hence we define ´ p sgn.d a C eb/ D p sgn.8e 3 a C 4e 2 c  d 2 /

if d if d

p p

a C eb ¤ 0 a C eb D 0

.

For u such that Q.u/ > 0, let

p R.u/ D beu3 C 2ceu2 C 3deu C 4e 3 C .4e 2 C du/ Q.u/.

(6.7)

Then, for sufficiently large u > 0 we have Q.u/ > 0 and sgn.R.u// D  . p p p Proof. If p d a C eb D 0 and 8e 3 a C 4e 2 c  d 2pD 0, then b D d a=e and c D 2e a C d 2 =.4e 2 /. Consequently, Q.u/ D . au2  .d=2e/up e/2 , which contradicts discr.Q/ ¤ 0. Therefore, at least one of d a C eb D 0 p the hypothesis 3 2 2 and 8e a C 4e c  dp D 0 is non-zero. p 3 Suppose first that d a C be ¤ 0. Then p limu!1 R.u/=u D d a C be, which proves our claim in this case. Next, let d a C be D 0. Then r r b b R.u/ 3de 4e 3 2 D .be C d a C a C . C   /u C .2ce C 4e C   / C C u2 u u u u2 p p As u tends to infinitypthe right-hand side tends to bd=.2 a/C.2ce C4e 2 a/C0C0, p which, in view of d a C be D 0, is equal to .8e 3 a C 4e 2 c  d 2 /=.2e/, hence of the same sign as  .

62

Chapter 6 Linear form for the quartic equation

Now, let

p c (6.8) x0 D 2e a C . 3 Following the notation in Chapters 3 and 4, we set f .X / D X 3 C AX C B and denote by e1 , e2 , e3 the roots of f .X /, enumerating them as explained on page 55. The following proposition describes the position of x0 relatively to e1 , e2 , e3 . p p Lemma 6.2. If d a C eb ¤ 0, then either x0 > e1 or e3 < x0 < e2 . If d a C eb D 0, then x0 D e1 and  D C1. p Proof. p It is easy to check that f .x0 / D x03 C Ax0 C B D .d a C be/2 . If d a C be ¤ 0, then f .x0 / > 0 and, consequently, either x0 > e1 , or (in the case that p e2 , e3 2 R) e3 < x0 < e2 . If d a C eb D 0, then x0 is a root of f .X / and we set f .X / D .X  x0 /h.X /, X CAp C x02 . In particular, e2 , e3 are the roots of h.X /. But where h.X / D X 2 C x0p 2 3 C 4ce 2  d 2 /=e; for the last equality we used h.x0 / D p3x0 C A D a .8e a p b D d a=e. By Lemma 6.1, 8e 3 a C 4ce 2  d 2 ¤ 0, hence x0 ¤ e2 , e3 and, if e2 , e3 2 R, consequently, x0 D e1 . Moreover, if e2 , e3 62 R, then h.x0 / > 0; and p then x0 > e2 , so that again h.x0 / > 0. Hence,  D sgn.8e 2 a C 4ce a C bd / D sgn.h.x0 // D C1. Lemma 6.3. Let u0  1 be such that for u  u0 the conclusion of Lemma 6.1 holds. For u  u0 define p cu2 C 3du C 6e 2 C 6e Q.u/ x.u/ D . (6.9) 3u2 Then, in the interval .u0 , C1/, x is strictly increasing or strictly decreasing, according to whether  D 1 or C1, respectively. Moreover, limu!C1 x.u/ D x0 . R.u/ D  3p and combine this with Proof. For the first claim, just observe that dx.u/ du u Q.u/ Lemma 6.1. The claim about the limit is obvious.

Now we are in a position prove a proposition which subsequently will help us to R C1 to pdw as a linear forms in elliptic logarithms. express the integral u Q.w/

Proposition 6.4. Let u0 be as in Lemma 6.3. Then, a constant u  u0 exists such that, for u  u we have Z C1 Z x.u/ dw dt D . (6.10) p p u x0 Q.w/ f .t / Proof. It is an elementary exercise to see with the aid of Lemmas 6.3 and 6.2 that, for u  u0 , the following cases are possible:

Chapter 6 Linear form for the quartic equation







63

p d a C be ¤ 0 and x0 > e1 . Then, x is strictly increasing when  D 1 and strictly decreasing when  D C1, with limiting value (in both cases) x0 , as u ! C1. Hence, for an appropriate u  u0 , it is indeed true that u  u ) x.u/ 2 .x0 , C1/  .e1 , C1/. p d a C be D 0 and x0 D e1 . Then, necessarily,  D C1, hence x is strictly decreasing with limiting value x0 , as u ! C1. Same conclusion as in the previous “bullet”. p d a C be ¤ 0 and x0 2 .e3 , e2 /. Then x is strictly decreasing when  D C1 and strictly decreasing when  D 1 with limiting value (in both cases) x0 2 .e3 , e2 /, as u ! C1. Therefore, for an appropriate u  u0 , it is true that u  u ) x.u/ 2 .x0 , e2 /  .e3 , e2 / if  D C1 and x.u/ 2 .e3 , x0 /  .e3 , e2 / if  D 1; hence, in both cases, x.u/ 2 .e3 , e2 /.

Let u be as in the above “bullets”. Then, for u  u0 , the interval .u, C1/ is mapped by x onto the open interval with end points x.u/ and x0 , which is a subset of .e1 , C1/ in the case of the first two “bullets” and a subset of .e3 , e2 / in the case of the last one. Therefore, in the right-hand side of (6.10)p we can make the change of variable t D x.w/ D .6ez C 3t d C ct 2 6e 2 /=.3t 2 /, z D Q.w/, so that dt D

.6e  dz C 3d  dw C 2cw  dw/w  2.6ez C 3dw C cw 2 C 6e 2 /  dw . 3w 3

By z 2 D Q.w/ we have dz D .4aw 3 C 3bw 2 C 2cw C d /=z  dw and, on replacing for dz in the last displayed equation, we obtain after some elementary calculations, dt D 

R.w/ dw bew 3 C 2cew 2 C 3dew C dwz C 4e 3 C 4e 2 z dw  D 3  , w3 z w z

where R is defined in (6.7). A symbolic computation shows that .R.w/=w 3 /2 D f .x.w// D fp .t / and, since w  u0 , we know that sgnR.w/ D  . Therefore, R.w/=w 3 D  f .t / and the last displayed equation becomes p

dt f .t /

D

dw dw , D p z Q.w/

which implies (6.10). Proposition 6.5. Let u be as in Proposition 6.4. Let P be a point of E, such that P C has real coordinates .u.P /, v.P // with u.P /  u and v.P /  0. Then, for the coordinates .x.P /, y.P // of P E we have x.P / D x.u.P // and y.P /  0. Proof. Let us p write for simplicity, u, v, x, y instead of u.P /, v.P /, x.P /, y.p/. Certainly, v D Q.u/ and, by (6.4), we have p 6e Q.u/ C cu2 C 3du C 6e 2 6ev C cu2 C 3du C 6e 2 D D x.u/ xD 3u2 3u2

64

Chapter 6 Linear form for the quartic equation

and y D  R.u/=u3 , where R.u/ is given in (6.7). In view of Lemma 6.1,  R.u/ > 0, as claimed. From now on, P will denote a point on E such that v.P / > 0 and u.P / is an integer  u , where u is as in Proposition 6.4. Also, we define P0 2 E by p (6.11) P0E D .x.P0 /, y.P0 // D .x0 ,  .be C d a// (cf. beginning of proof of Lemma 6.2). We will use, implicitly, Proposition 6.5 as well as the conclusions of the “bullets” in the proof of Proposition 6.4. We will also use the following lemma. Lemma 6.6. If e2 , e3 2 R and e3  x  e2 , then Z C1 Z x dt dt D , p p e2 x0 f .t / f .t / where .e1  e2 /.e2  e3 / x 0 D e2 C 2 Œe1 , C1/ e2  x In particular, in the case that x D x.P / 2 Œe3 , e2 , we have x 0 D x.P C Q2 /, where Q2 is the point of E with Q2E D .e2 , 0/ (see (3.7)). Proof. For t 2 Œx, e2 / we make the change of variable .e1  e2 /.e2  e3 / .e1  e2 /.e2  e3 / t D e2 C so that t 0 D e2 C . 0 e2  t e2  t The function t 7! t 0 is strictly increasing in the interval Œx, e2 /, with values in Œx 0 , C1/. A simple calculation now proves the equality of the integrals, as stated in the announcement of the lemma. Concerning the identity x 0 D x.P C Q2 /, its proof is a matter of an easy (even by hand) symbolic calculation. p Now we proceed further, supposing first that eb C d a ¤ 0. According to Lemma 6.2, either x0 > e1 , or e2 , e3 2 R and x0 2 .e3 , e2 /. If x0 > e1 , then (first “bullet” in the proof of Proposition 6.4), x.P / D x.u.P // > e1 , so that we can write Z C1 Z C1 Z C1 Z C1 Z x.P / dt dt dt dt dt p p p p p D  D  . f .t / f .t / f .t / f .t / f .t / x0 x0 x.P / x.P0 / x.P / (6.12) If e2 , e3 2 R and e3 < x0 < e2 , then, for t 2 Œe3 , e2 / we write t 0 D e2 C

.e1  e2 /.e2  e3 / e2  t

Chapter 6 Linear form for the quartic equation

65

(cf. proof of Lemma 6.6). By the third “bullet” in the proof of Proposition 6.4, x.P / D x.u.P // 2 .e3 , e2 / and, using Lemma 6.6, we compute Z e2 Z e2 Z x.P / dt dt dt D  p p p x0 x0 x.P / f .t / f .t / f .t / Z C1 Z C1 dt dt D  p p 0 x0 x.P /0 f .t / f .t / Z C1 Z C1 dt dt D  . (6.13) p p x.P0 CQ2 / x.P CQ2 / f .t / f .t / p Next, suppose that be C d a D 0. Then, by Lemma 6.2, x0 D e1 ,  D C1 and the second “bullet” in the proof of Proposition 6.4 x.P / D x.u.P // > e1 . Therefore we can write Z C1 Z C1 Z x.P / dt dt dt D  . (6.14) p p p x0 e1 x.P / f .t / f .t / f .t / All three integrals in the left-hand sides of relations (6.12)–(6.14) are, by Proposition 6.4, equal to Z C1 dw  p u.P / Q.w/ and, moreover, in (6.14) we know that  D C1. On the other hand, we look at the right-hand sides of relations (6.12)–(6.14). For both relations (6.12) and (6.13), the right-hand side is equal to 12 .l.P0 /"P0 Cl.P /"P /, in view of the relation (3.32). Further, in this case, by the definition of P0 , we have y.P0 / > 0, hence "P0 D 1. Also, since u.P / > u0 and v.P / > 0, Proposition 6.5 asserts that y.P / > 0, hence "P D 1 and, consequently, the right-hand side of both (6.12) and (6.13) is equal to  12 l.P0 / C 12 l.P /. In the case of the relation (6.14), the first integral is equal to !1 , in view of (3.18). Then, using (3.32) as before, we conclude that the right-hand side of (6.14) is !1 C 12 l.P /. A straightforward combination of our conclusions above proves the following proposition. Proposition 6.7. Let P 2 E be such that u.P /  u and v.P / > 0. 

If either x0 > e1 , or e2 , e3 2 R and e3 < x0 < e2 , then Z C1  dw D .l.P /  l.P0 //. p 2 u.P / Q.w/



If x0 D e1 (so that, necessarily,  D C1), then Z C1 1 dw D !1 C l.P /. p 2 u.P / Q.w/

66

Chapter 6 Linear form for the quartic equation

Now we are ready for the main result of this chapter. Theorem 6.8. Let E be the elliptic curve represented by the model C in (6.1). Let E : y 2 D x 3 C Ax C B D f .x/ be the short Weierstrass model with A, B defined in (6.3) and consider that, in Chapter 4, equation (4.1) as well as everything else, in particular relations (4.4)–(4.8), refers to this particular model E. Choose a number u following the guidelines in the proof of Proposition 6.4 and then choose u  u and c7  1 such that, jx.u/j  c7

for all u  u .

Assume that P 2 E is such that P C D .u.P /, v.P // has integer coordinates with v.P / > 0 and u.P /  u and write P E as in (4.2). With l.P / as in (4.6), define L.P / by ´ l.P /  l.P0 / if either x0 > e1 , or e2 , e3 2 R and e3 < x0 < e2 L.P / D l.P / C 2!1 if x0 D e1 , where e1 , e2 , e3 are the roots of f .X /, as described in page 55, !1 is the least positive real period of the Weierstrass } function associated with the short Weierstrass model E, and P0 is the point of E with P0E as in (6.11). Then, 4 jL.P /j  p  exp. 12 log.3c7 / C   M 2 /, a

(6.15)

where  and are defined in Propositions 2.6.3 and 2.6.2, respectively. Proof. First we show that there exist pairs .u , c7 / as in the announcement of the theorem. Indeed, let any u  u . Then u  u0 (see Lemma 6.3) and, consequently, Q.u/ > 0. Then, by (6.9), p p jcu2 C 3du C 6e 2 C 6e Q.u/j jcju2 C 3jd ju C 6e 2 C 6e Q.u/ jx.u/j D  . 3u2 3u2 Obviously, there exists u  u such that, if u  u , then 0 < Q.u/  2au4 and, consequently, by p the above displayed relation, jx.u/j  c7 , where c7 D max¹1, .jcj C 3jd j C 6e 2 C 6e 2a/=3º; we ask the reader to have a look at the remark after the proof. Now, consider a point P as in the announcement of the theorem. By the definition of L.P / and Proposition 6.7 we have Z C1 4 dw  p  u.P /1 , (6.16) jL.P /j D 2 p u.P / Q.w/ a where one can prove the inequality on the right exactly as we did for the upper bound of the integral in Proposition 5.4.

Chapter 6 Linear form for the quartic equation

67

p Since v.P / > 0, we have v.P / D Q.u.P // and then, in view of (6.4) and (6.9), x.P / D x.u.P //. Let x.P / D m=n, where m, n are relatively prime integers. But also, x.P / D .x.u.P //  3u.P /2 /=.3u.P /2 / where the numerator and denominator are integers; actually, the numerator is equal to c u.P /2 C3d u.P /C6e 2 C6e v.P /. Therefore, jmj  3jx.u.P //j  u.P /2  3c7  u.P /2 , jnj  3u.P /2  3c7  u.P /2 and, consequently, for the logarithmic height of the rational number x.P / we have (see (2.21)) h.x.P // D log max¹jmj, jnjº  log.3c7  u.P /2 / D log.3c7 / C 2 log u.P /. (6.17) Then,

1 1 log.3c7 /  h.x.P // (by (6.17)/ 2 2 1 O / (by Proposition 2.6.3)  log.3c7 / C   h.P 2 1 (by Proposition 2.6.2).  log.3c7 / C   M 2 2 Now, a straightforward combination of the last inequality with (6.16) proves (6.15). log.u.P //1 

Remark. The value of c7 given in the proof was obtained under the assumption that u is so large that Q.u/  2au4 . Obviously, this choice is arbitrary; we could have chosen any constant > 1 and then assumed that u is so large that Q.u/  constant  u4 . Solutions .u, v/ with u < 0. On the first glance, Theorem 6.8 seems to treat integer solutions .u, v/ to (6.1) with “large” positive u, only. However, as we will show below, with a very little extra effort we can treat all integer solutions .u, v/ with “large” juj and obtain an exactly analogous upper bound in which the assumption concerning the “large” unknown point P C D .u.P /, v.P // is that “ju.P /j is sufficiently large” rather than “u.P / is sufficiently large”. It is clear what one should do; it suffices to replace the coefficients a, b, c, d , e of the given equation by a, b, c, d , e, respectively, and consider the equation v 2 D au4  bu3 C cu2  du C e 2 in integer .u, v/ with sufficiently large u > 0. How does the replacement of .a, b, c, d , e/ by .a, b, c, d , e/ affect the crucial “quantities” (= parameters and auxiliary functions) involved in the discussion so far? This is shown in Table 6.1. The symbols of the “quantities” involved in our discussion are on the left, while on the right the analogous symbols are listed with a bar  on them, denoting the analogous “quantities” which result when the coefficients b, d are replaced by b, d , respectively. We avoid the bar indication if the replacement does not affect the “quantity”.

68

Chapter 6 Linear form for the quartic equation

Table 6.1. Parameters and auxiliary functions for the solution of the quartic elliptic equation. 0 10 Q.u/ D au4 C bu3 C cu2 C du C e 2

A, B (see (6.3)) x0 (see (6.8))

Q.u/ D au4  bu3 C cu2  du C e 2 p ´  if d a C be ¤ 0 p D  if d a C be D 0 A, B x0

R.u/ D beu3 C 2ceu2 C 3deu C 4e 3 C.4e 2 C du/  Q.u/1=2

R.u/ D beu3 C 2ceu2  3deu C 4e 3 C.4e 2  du/  .Q.u//1=2



x.u/ D

(see Lemma 6.1)

cu2 C 3du C 6e 2 C 6e  .Q.u//1=2 3u2

x.u/ D

cu2  3du C 6e 2 C 6e  .Q.u//1=2 3u2

Choose u0  1 so that

Choose u0  1 so that

u  u0 ) Q.u/ > 0 & sgn.R.u// D 

u  u0 ) Q.u/ > 0 & sgn.R.u// D 

Choose u  u0 so that In the case that x0  e1 : u > u ) x.u/ > e1

Choose u   u0 so that In the case that x0  e1 : u > u  ) x.u/ > e1

In the case that e3 < x0 < e2 : u > u ) e3 < x.u/ < e2

In the case that e3 < x0 < e2 : u > u  ) e3 < x.u/ < e2

Choose u  u and c7 > 0 so that jx.u/j  c7 for all u  u

Choose u   u  and c 7 > 0 so that jx.u/j  c 7 for all u  u  p ´ P0E if d a C be ¤ 0 E p P0 D if d a C be D 0 P0E p ´ l.P0 / if d a C be ¤ 0 p l.P 0 / D if d a C be D 0 l.P0 /

P0E D .x0 ,  .be C d

p

a//

l.P0 / L.P / 

if either x0 > e1 or e2 , e3 2 R and x0 2 .e3 , e2 / then L.P / D l.P /  l.P0 / L.P / D l.P /  l.P0 /



if x0 D e1 then L.P / D l.P / C 2!1

L.P / 

if either x0 > e1 or e2 , e3 2 R and x0 2 .e3 , e2 / then p ´ l.P / C l.P0 / if d a C be ¤ 0 p L.P / D l.P /  l.P0 / if d a C be D 0  if x D e then 0 1 L.P / D l.P / C 2!1

To continue with the resolution of the quartic elliptic equation, one can safely avoid Chapters 7 and 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.4.

Chapter 7

Linear form for simultaneous Pell equations

Let A1 , A2 be given non-zero integers and let D1 , D2 be given positive non-square integers. In this chapter we will deal with the system U 2  D1 V 2 D A1 ,

W 2  D2 V 2 D A2

(7.1)

of simultaneous Pell equations, where .U , V , W / are positive integers, under the assumption that we know a rational solution .U0 , V0 , W0 / with V0 > 0 and U0 , W0  0, not both zero. This assumption is not really restrictive in practice. All known examples possess at least one small solution, easily found by a direct search; see e.g. the table in [60, Appendix]. An alternative method to discover a rational, or even integral, solution .U0 , V0 , W0 / is to consider the equation .D1 V 2 C A1 /.D2 V 2 C A2 / D Z 2 (Z D U W ) and search for rational solutions. For example, if we take .D1 , A1 / D .7, 2/ and .D2 , A2 / D .32, 23/ (an example from Z. Y. Chen, mentioned in [60, Appendix]), then the MAGMA routine Points applied to the corresponding curve1 immediately returns the points .V , Z/ D .1, 9/, .271, 1099161/, which furnish the solutions .U , V , W / D .3, 1, 3/, .717, 271, 1533/. In [60] an alternative method for the explicit resolution of (7.1) is developed, which requires the solution of a number of quartic Thue equations; moreover, that paper contains a rich bibliography on simultaneous Pell equations, to which we refer the interested reader. Anyway, choosing between Ellog and Thue equations for the resolution of a certain Diophantine problem does not admit a general answer; this issue is discussed in the beautiful paper [58]. Now, let us come back to (7.1). To each solution .U , V , W / we will associate a point P on a certain short Weierstrass model, along with a linear form L.P / and then we will prove Theorem 7.1, which will provide us with an explicit upper bound for jL.P /j (cf. page 56 immediately after (4.8)). This chapter’s exposition is based on the author’s paper [63]. Throughout the chapter we keep the notations, assumptions and results of Chapter 4. Clearly, for a solution .U0 , V0 , W0 / as above, p p (7.2) sgn.U0  V0 D1 / D sgn.A1 /, sgn.W0  V0 D2 / D sgn.A2 /. Moreover, we will assume that A1 D2  A2 D1 ¤ 0, otherwise the system is rather trivially solved. Indeed, if A1 D2  A2 D1 D 0, then (7.1) implies D2 U 2 D D1 V 2 , so that A2 =A1 D D2 =D1 D q 2 for some q 2 Q and, consequently, the solutions 1

Viewed as an “hyperelliptic curve” in MAGMA’s language.

70

Chapter 7 Linear form for simultaneous Pell equations

.U , V , W / of (7.1) are given by .U , V , qU /, where .U , V / is a solution of the first equation, satisfying qU 2 Z. In view of symmetry of the pairs .A1 , D1 / and .A2 , D2 /, we may assume, without loss of generality, that A1 D2  A2 D1 > 0. If in this relation we replace A1 by U02  D1 V02 and A2 by W02  D1 V02 we see that p p (7.3) U0 D2  W0 D1 > 0. Now let .U , V , W / be any solution (not necessarily integral) to (7.1), such that .V : W / ¤ ˙.V0 : W0 /. Combining the two equations (7.1) we obtain A2 U 2  A1 W 2 C .A1 D2  A2 D1 /V 2 D 0. We put uD

U0 V  V0 U , W0 V C V0 W

(7.4)

(7.5)

so that

V0 .U C uW / . (7.6) U0  uW0 Then we substitute for V in (7.4) to obtain a quadratic equation in U=W . One solution of that equation is, obviously, U=W D W0 =U0 , hence the other solution is V D

A2 U0 u2  2A1 W0 u C A1 U0 U . D W A2 W0 u2  2A2 U0 u C A1 W0

(7.7)

Then, using (7.6) we obtain .A2 u2  A1 /V0 V . D W  A2 W0 u2 C 2A2 U0 u  A1 W0 V We have 1  D2 . 70W /2 D

A2 , 80W 2

(7.8)

hence, (7.8) and the relations

U02  D1 V02 D A1 ,

W02  D2 V02 D A2 ,

(7.9)

imply that def

v 2 D au4 C bu3 C cu2 C du C e 2 D Q.u/, where

A2 W0 u2  2A2 U0 u C A1 W0 , W p  D sgn.A1 / D sgn.U0  V0 D1 /,

vD

(7.10) (7.11) (7.12)

.a, b, c, d / D .A22 , 4A2 U0 W0 , 4A1 W02  2A1 A2 C 4A2 U02 , 4A1 U0 W0 , jA1 j/. (7.13) Equation (7.10) has been studied in Chapter 6. There we focused our interest on integer solutions .u, v/ with arbitrarily large u > 0, whilst now the u and v in (7.5) and (7.11),

71

Chapter 7 Linear form for simultaneous Pell equations

respectively, are certainly rationals but, in general, not integral and, moreover, as we will see, u belongs to a finite interval instead of approaching infinity. Nevertheless, we will exploit part of the results of our study in Chapter 6 in order to solve our present problem. In (7.11) we replace u by its expression in (7.5) in order to express v as a rational function of U , V , W (we also take into account (7.9)): A2 U0 U C .D2 A1  D1 A2 /V0 V C A1 W0 W . (7.14) v D 2V02 .V0 W C W0 V /2 Conversely, we need also express U , V , W as rational functions of u, v. By the definition of v, A2 W0 u2  2A2 U0 u C A1 W0 W D (7.15) v and this, combined with (7.8) and (7.7) gives A2 U0 u2  2A1 W0 u C A1 U0 .A2 u2 C A1 /V0 , U D . (7.16) v v Now, using (6.4), (6.5) and (6.6), we establish a birational transformation between (7.10) and def (7.17) E : y 2 D x 3 C Ax C B D f .x/, V D

with A and B expressed by the formulas (6.3) as functions of a, b, c, d , e taken from (7.13). R As in Chapter 6, we will express an elliptic integral pdt , with appropriate upf .t/

per and lower limits, in terms of another integral in which the variable V is directly involved. This will be accomplished with the aid of a function x.V /, defined on an interval ŒV  , C1/, for a sufficiently large positive V  . Below we proceed to construct this function. V0 , W0 / be any positive real solution to (7.1), so that U D p Let .U , V , W / ¤ .U0 ,p A1 C D1 V 2 and W D A2 C D2 V 2 . Then (7.5) and (7.11) suggest that we define the functions p U0 V  V0 A1 C D1 V 2 (7.18) V 7! u.V / D p W0 V C V0 A2 C D2 V 2 and A2 W0  u.V /2  2A2 U0  u.V / C A1 W0 p , (7.19) V 7! v.V / D  A2 C D2 V 2 p which satisfypthe relation v.V /2 D Q.u.V //. The substitutions U A1 C D1 V 2 2 and W A2 C D2 V in (7.5) makes u D u.V /, v D v.V / (by (7.15)), and consequently, in view of (7.14), p p 2 2 2 A2 U0 A1 C D1 V C .A1 D2  A2 D1 /V0 V C A1 W0 A2 C D2 V v.V / D 2V0 p . .W0 V C V0 A2 C D2 V 2 /2 (7.20)

72

Chapter 7 Linear form for simultaneous Pell equations

After some standard computations, in which use is made of the definition of  (see (7.12)), as well as of the relations (7.2) and (7.3), we verify the following (dashes denote derivatives): p U0  V0 D1 def p . (a) u1 D lim u.V / D V !C1 W0 C V0 D2 p p p . D2 U0 C D1 W0 /.U0  V0 D1 /V0 3 0 p p . (b) lim V  u .V / D V !C1 D1 D2 .W0 C V0 D2 / (c)

lim v.V / D 0.

V !C1

p p p 2 jU  V  2V D j.U D C W D1 / 0 0 1 0 2 0 0 . (d) lim V 2  v0 .V / D p V !C1 W0 C V0 D2 (e) Q.u1 / D 0.

p p p p  8V03 D1 D2 .U0  V0 D1 /.U0 D2 C W0 D1 / (f) Q .u1 / D p . W0 C V0 D2 We note first that, by (e), the polynomial Q has four real roots which, p in view p of (a), are actually the algebraic conjugates of u1 in the quartic field Q. D1 , D2 /; temporarily, let us denote these roots by r4 < r3 < r2 < r1 . If  D 1, then, by (f), Q is strictly increasing in an interval with u1 as its centre, therefore, u1 D r1 or r3 and Q.u/ > 0 in an interval I with u1 as left end point and right end point appropriately close to u1 (or even C1 if u1 D r1 ). If  D C1, then an analogous argument shows that u1 D r2 or r4 and Q.u/ > 0 in an interval I with u1 as right end point and left end point appropriately close to u1 (or even 1 if u1 D r4 ). Then, with the interval I as above, it is clear that p Q.u/ C cu2 C 3du C 6e 2 def 6e , (7.21) I 3 u 7! x.u/ D 3u2 p ebu3 C 2ecu2 C 3edu C 4e 3 C .du C 4e 2 / Q.u/ def I 3 u 7! y.u/ D  (7.22) u3 are meaningful as real-valued functions; note that the formula for x.u/ is the same as that in (6.9), but the domains of definition differ: There, the domain is a neighbourhood of C1, here the domain is thepinterval I . We also remark that, if in the right-hand sides of (6.4) we replace v by Q.u/, then we obtain the right-hand sides of (7.21) and (7.22), respectively, except that  is defined differently. Note, however, that a different choice of  in (6.4), (6.5) and (6.6) does not affect the fact that these relations establish a birational transformation between v 2 D Q.u/ and y 2 D x 3 C Ax C B; cf. the comment just before the announcement of Lemma 6.1. 0

73

Chapter 7 Linear form for simultaneous Pell equations

Next we note that, in view of (c) and (d) on page 72, if V is sufficiently large, then v.V / > 0. Having in mind the above remarks, let .u, v/ D .u.V /, v.V //. In view of (a) and (b) on page 72, if V is sufficiently large, then u D u.V / 2 I and v Dpv.V / > 0. Since v 2 D v.V /2 D Q.u.V // D Q.u/, we conclude then that v D Q.u/ p and, consequently, from (7.21) and (6.4) we see that x.u/ D X .u, v/ p D X .u, Q.u//. Similarly, (7.22) and (6.4) imply that y.u/ D Y.u, v/ D Y.u, Q.u//. Therefore, for sufficiently large V , the functions def

V 7! x.V / D .x ı u/.V /,

def

V 7! y.V / D .y ı u/.V /

are meaningful as real-valued functions and .x.V /, y.V // D .X .u.V /, v.V //, Y.u.V /, v.V ///,

v.V / D

(7.23)

p Q.u.V // ;

in particular, .x.V /, y.V // 2 E.R/; just remember that, if u, v 2 R and v 2 D Q.u/, then .x, y/ D .X .u, v/, Y.u, v// is a point of E.R/. Summing up: Any integer solution .U , V , W / to (7.1) with V > 0 and U , W  0 corresponds to a point .x, y/ D .x.V /, y.V // 2 E.Q/. Conversely, from .x, y/ we can recover the solution .U , V , W /, first, by obtaining .u, v/ using (6.5) and (6.6), and then by using (7.16) and (7.15). Next note that we can also define R.u/ by (6.7) and we check that p p p p p 8V03 .U0 C V0 D1 /jU0  V0 D1 j3 .U0 D2 C W0 D1 / D1 D2 p , R.u1 / D .W0 C V0 D2 /2 whence R.u/ > 0. But then, similary to the proof of Lemma 6.3, we see that the function x is strictly decreasing in the interval I . On the other hand, by (b) on page 72, for sufficiently large V , u is strictly increasing or decreasing, according to whether  D C1 or  D 1, respectively. Therefore, we finally conclude that, for sufficiently large V , the function V 7! x.V / is strictly increasing or decreasing according to whether  D 1 or  D C1, respectively. Further, we compute def

x0 D D

and

lim x.V / D X .u1 ,

V !C1 4V02 

3

p Q.u1 //

(7.24)

p D2 U02 C D1 D2 V02 C D1 W02 C 3 D1 D2 U0 V0 p p  C 3D1 D2 V0 W0 C 3 D1 D2 U0 W0 > 0

p lim y.V / D Y.u1 , Q.u1 // (7.25) V !C1 p p p p p D 8V03 D1 D2 .W0 D1 C U0 D2 /.U0 C V0 D1 /.W0 C V0 D2 / > 0. def

y0 D

74

Chapter 7 Linear form for simultaneous Pell equations

p Note that .x0 , y0 / is a point of E.R/; more precisely, a point of E.Q. D1 D2 //. Indeed, look at (7.24) and (7.25) and, once again, remember that any .x, y/ D .X .u, v/, Y.u, v// with real u, v and v 2 D Q.u/ is a point of E.R/. The point p def P0E D .x0 , y0 / 2 E.Q. D1 D2 // (7.26) plays a role in our method; see Theorem 7.1 below. In the applications we will find important the property that 2P0E 2 E.Q/; more specifically, 2P0E D . 43 .D1 D2 V02 C D1 W02 C D2 U02 /V02 , 8D1 D2 U0 W0 V04 /.

(7.27)

Coming back to x.V / and y.V /, we conclude that, according to our above discussion, if V 2 .V  , C1/ for a sufficiently large V  , then x.V / 2 I.x0 /, where I.x0 / is an interval .x1 , x0 / or .x0 , x1 /, according to whether  D C1 or  D 1,  respectively, for an p appropriate x1 . Moreover, for V 2 .V , C1/ we have y.V / > 0, whence y.V / D f .x.V //. Rx Now we focus our interest on the integral x 0 pdt when x belongs to the interval f .t/

I.x0 / as above. According to our previous discussion, the interval with end points x (included) and x0 (excluded) is the image under the function x D x ı u of an interval the change of ŒV , C1/, where x D x.V /. Therefore, for t 2 Œx, x0 / we can make p variable t D x.T / D x.u.T //, where T 2 ŒV , C1/. Then y.T / D f .x.T // D p f .t / and dx du dt D  . dT du dT From (7.21) we calculate p eb  u3 C 2ec  u2 C 3ed  u C 4e 3 C .d  u C 4e 2 / Q.u/ dx p D du u3 Q.u/ p f .t / y y.u/ D  D  . D (by (7.22))   p v v Q.u/ From (7.18) we compute p p V0 .A2 U0 A1 C D1 T 2 C .A1 D2  A2 D1 /V0 T C A1 W0 A2 C D2 T 2 du p p p D dT .V0 A2 C D2 T 2 C W0 T /2 A1 C D1 T 2 A2 C D2 T 2  v . D (by (7.20)) p 2V0 .A1 C D1 T 2 /.A2 C D2 T 2 / On combining the above relations we obtain

p f .t / 1 dt p D dT 2V0 .A1 C D1 T 2 /.A2 C D2 T 2 /

and, consequently, Z C1 Z x0 1 dt dT D . p p 2V0 V x.V / f .t / .A1 C D1 T 2 /.A2 C D2 T 2 /

(7.28)

Chapter 7 Linear form for simultaneous Pell equations

75

Now we examine more closely the short Weierstrass model E defined by (7.17). First we note that the polynomial f .X / has the following three rational roots: 4 4 1 D .D2 X02  2D1 D2 Y02 C D1 Z02 /Y02 D .A1 D2 C A2 D1 /Y02 3 3 4 4 2 D .2D2 X02 C D1 D2 Y02 C D1 Z02 /Y02 D .A2 D1  2A1 D2 /Y02 3 3 4 4 3 D .D2 X02 C D1 D2 Y02  2D1 Z02 /Y02 D .A1 D2  2A2 D1 /Y02 . 3 3 Note that, with our ordinary notation, the roots of f .X / are denoted by e1 , e2 , e3 , where e1 > e2 > e3 ; therefore, .e1 e2 e3 / is a permutation of . 1 2 3 /. Now we consider the point P0 with P0E D .x0 , y0 / 2 E.R/ defined in (7.26). We easily check that x0  i > 0 for i D 1, 2, 3, therefore, P0E 2 E0 .R/ and, consequently, according to our comment immediately after (7.25), we may assume that, if V is sufficiently large, then the point PV , which is defined by P E D .x.V /, y.V //, belongs to E0 .R/ and y.V / > 0. Then, we can write the left-hand side of (7.28) as follows: Z C1 Z C1 Z x0 dt dt dt D  D 2.l.P /  l.P0 // (by (3.32)). p p p x.V / x0 x.V / f .t / f .t / f .t / Easily, the right-hand side of (7.28) is absolutely less than apconstant times V , provided that V is not “very small”. For example, if V  maxp iD1,2 ¹ 2jAi j=Di º, then the right1 hand side of (7.28) is absolutely less than V =.V0 D1 D2 / and, consequently, jl.P /  l.P0 /j
0 and U0 , W0  0, U0 C W0 > 0, be a known rational solution of the system (7.1). Let .U , Vp, W / be an integer solution p of the system (7.1) with U , V , W positive, so that U D A1 C D1 V 2 , W D A2 C D2 V 2 and, consequently, any function of .U , V , W / can be viewed as a function of V only. Assume that V is sufficiently large, so that every condition below is fulfilled: p  For i D 1, 2 it is true that Ai C Di V 2 is bounded by a constant times V .2 



2

u.V / (cf. (7.18)) and v.V / (cf. (7.19)) satisfy Q.u.V // > 0 and v.V / > 0, respectively. V belongs to an infinite interval .V  , C1/ in which the functions x and y, defined in (7.23), have the following properties: The function x is strictly monotonous, x.V / is larger than the maximum of the i ’s displayed at page 75 and y.V / > 0. p p For example, this constant can be chosen to be 2 Di if Ai > 0 and Di if Ai < 0.

76

Chapter 7 Linear form for simultaneous Pell equations

Let E be the Weierstrass model defined in (7.17) and consider that, everything in Chapter 4, in particular the relations (4.1) and (4.4)–(4.8), refers to this particular model E. Consider also the points P E D .x.V /, y.V // 2 E.Q/ and P0E defined in (7.26). Express P E as in (4.2) and l.P / by (4.6), and set L.P / D l.P /  l.P0 /. Then, jL.P /j 

1 p  exp. 12 c8 C   M 2 /, 2V0 D1 D2

(7.30)

where 12 0 and  are as in Propositions 2.6.2 and 2.6.3, respectively, and c8 is a positive constant which can be explicitly calculated as explained below, in the paragraph immediately before relation (7.32). Proof. Firstly, by what we have already seen on pages 73 and 74, .x.V /, y.V // and .x0 , y0 / are, indeed, points of E.R/. Further, we saw on page 75 that .x0 , y0 / 2 E0 .R/ and, if V is sufficiently large, then also .x.V /, y.V // belongs to E0 .R/. More prep cisely, .x0 , y0 / 2 E0 .Q. D1 D2 // and, since U , V , W are integers, .x.V /, y.V // 2 E0 .Q/. Next, a quick look at our previous discussion after page 71 and before the relation (7.29) easily reminds that, indeed, for sufficiently large V all conditions marked by “bullets” are fulfilled, so we proceed to the main part of the proof, noting the following: If u and v are given by (7.5) and (7.14), respectively, then x.V / D

6ev C 6u2 C 3du C 6e 2 . 3u2

(7.31)

The right-hand side can be written as an element of Q.U , V , W /, the numerator and denominator of which are both of total degree 2. On multiplying numerator and denominator by `2 , where ` is the least common multiple of the denominators of U0 , V0 , W0 , we obtain a relation of the form x.V / D P1 .U , V , W /=P2 .U , V , W /, where P1 , P2 are polynomials in U , V , W with explicit integer p coefficients belong, D , .`U /, .`V /, .`W /. Then, since U D A1 C D1 V 2 and W D ing to ZŒD 1 2 0 0 0 p A2 C D2 V 2 , it is clear that, for i D 1, 2, we have jPi .U , V , W /j  consti  V 2 , where consti is an explicitly computable positive real number.3 Let now x.V / D m=n, where m, n are relatively prime integers, so that h.x.V // D log max¹jmj, jnjº. We have m=n D x.V / D P1 .U , V , W /=P2 .U , V , W / and the right-most quotient has integer numerator and denominator. Therefore, max¹jmj, jnjº  max¹P1 .U , V , W /, P2 .U , V , W /º  max¹const1 , const2 º  V 2 , from which it becomes clear that h.x.V //  c8 C 2 log V 3

hence

1 1 log V 1  c8  h.x.V // 2 2

Here we need the requirement set by the first “bullet” in the announcement of the theorem.

(7.32)

Chapter 7 Linear form for simultaneous Pell equations

77

for some explicitly computable positive constant c8 ; in the notation of the previous lines, c8 D log max¹const1 , const2 º. O /  M 2 and O /  1 h.x.V //   ; by Proposition 2.6.2, h.P By Proposition 2.6.3, h.P 2 combining these with (7.32) gives O /  1 c8 C   M 2 . log V 1  12 c8  12 h.x.V //  12 c8 C   h.P 2

p By the definition of L.P / and (7.29) we have L.P / < V 1 =.2V0 D1 D2 / and, combining this with the last displayed inequality, gives the inequality (7.30). To continue with the resolution of the simultaneous Pell equations, one can safely avoid Chapter 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.5.

Chapter 8

Linear form for the general elliptic equation

In this chapter we treat general elliptic equations, C : g.u, v/ D 0,

(8.1)

where g is a polynomial with coefficients in Z, irreducible over Q and the curve represented by the model (8.1) is of genus one and possesses a non-singular point with rational coordinates. Throughout the chapter we keep the notations, assumptions and results of Chapter 4. This chapter is based mainly on the joint work [56] of R. Stroeker and the author. It is well known that the equation (8.1) has finitely many solutions (see, for example, [2] or [41]), but here, in analogy to Chapters 5, 6 and 7, we intend to prove Theorem 8.7.2, which gives an explicit upper bound for jL.P /j, where L.P / is an appropriate linear form in elliptic logarithms; cf. page 56 immediately after (4.8). Without loss of generality we assume that degu g  degv g D n. We will make the further assumption that C.R/ possesses infinite (= unbounded) branches along both the u-direction and the v-direction. This is very natural to assume, for, otherwise, the problem of solving g D 0 in integers is, obviously, trivial. Before proceeding to the study of (8.1), we collect all the assumptions that we set so far, which will be implicitly understood throughout the whole chapter. 

g is a polynomial with coefficients in Z, irreducible over Q.



The curve represented by the model (8.1) is of genus one and a non-singular point with rational coordinates is known.



degu g  degv g D n.



C.R/ possesses infinite branches along both the u-direction and the v- direction.

8.1 A short Weierstrass model As in previous chapters, a basic tool of the Ellog is the use of a short Weierstrass model birationally equivalent to the model C with which we started. Proposition 8.1.1. The model C is birationally equivalent over Q to a model E : y 2 D f .x/ D x 3 C Ax C B,

A, B 2 Q.

(8.2)

The birational transformation, as well as A and B, can be explicitly constructed.

79

Section 8.1 A short Weierstrass model

Proof. We refer to van Hoeij’s [22], where an algorithm is presented for calculating the birational transformation between C and a short Weierstrass model E, provided that a non-singular point .u0 , v0 / of C with algebraic coordinates is known. According to Section 3.2 of that paper, the coefficients of the transformation which is constructed, as well as the coefficients of the Weierstrass model E, belong to the field Q.u0 , v0 /; this is even more clear if one looks at van Hoeij’s papers [20] and [21] on which the algorithm of [22] is based. Since we have assumed that a non-singular point of C with rational coordinates is known, it follows that, if we insert this particular point in van Hoeij’s algorithm [22], we will get a birational transformation with rational coefficients between C and a short Weierstrass model E which will be defined over Q. The birational transformation of Proposition 8.1.1 will be denoted by C 3 .u, v/

.X ,Y/ !  .U,V/

.x, y/ 2 E ,

(8.3)

meaning that u.P / D U.x.P /, y.P //,

v.P / D V.x.P /, y.P //

x.P / D X .u.P /, v.P //,

y.P / D Y.u.P /, v.P //,

in accordance with Section 1.1. ExampleStep 1 . In this chapter, parallel to the general exposition, we will discuss the methodical solution in integers of the equation g.u, v/ D 0, where g.u, v/ D 3v 5 C 3uv 3  271uv  3u2 .

(8.4)

With the aid of MAPLE we find out that the genus of the curve C : g D 0 is one. Obviously, u D .0, 0/ is a point, so that C is a model of an elliptic curve over Q. The MAPLE implementation of van Hoeij’s algorithm [22] gives the birational transformation (8.3) between C and the model 120408061761 . (8.5) E : y 2 D f .x/ D x 3  17846163x C 4 The functions X and Y are the following:  X .u, v/ D 2710u  9225uv 3  43821uv  5904u2 C 21867uv 2 C 726v 4 C 2169u3  C 8883u2 v 2  48429vu2 C 2349v 4 u/=u.u  1/2  Y.u, v/ D 79947u4 C 14069376vu3  829440u3 v 2 C 1802709u3 C 137565u2 6952716v 3 u2 C 140667858vu2  39726990u2 v 2  1224558v 4 u2  22481070uv 2  4683906v 4 u C 11958534uv 3 C 34375266uv  C 81029u  395286v 4 /=.u.u  1/3 ,

(8.6)

80

Chapter 8 Linear form for the general elliptic equation

and the inverse functions1 are  U.x, y/ D 2x 5 C 776970x 4 C 6389652843x 3 C 2050yx 3  11183395769802x 2 C 126072288yx 2  123607085812402995x  199252067022xy C 268148092563378425394  930481914765852y/=2.x C 4365/2 .x  2169/3



V.u, v/ D  3.x 2  5121x  3270699 C 242y/=.x C 4365/.x  2169/.

(8.7)

Example continued on page 82.

8.2 Puiseux series Our intuition suggests that it should be possible to “solve for v” the equation g.u, v/ D 0 over the real numbers. Actually this is guaranteed by the implicit function theorem. Since, in our case, the real function .u, v/ 7! g.u, v/ is very special, namely, a polynomial function, we can say much more about the solutions v D v.u/, which can be expressed in a very concrete way by means of the Puiseux series. In general, let K be a subfield of the complex numbers and a 2P K. Roughly speaking, we obtain a Puiseux k series around a if in a usual power series C1 kDm qk .X  a/ (m 2 Z may be nega1= for some positive integer . For a systematic tive) we replace X  a by .X  a/ presentation of the relevant theory we refer to [66, Chapter IV] and/or [4, Chapter II];2 useful references are also [67, §3], [68], [69, §2], [70, §3]. We collect below all the information that is necessary for the purposes of the present chapter. Fact 8.2.1. (a) There is a finite Galois extension K=Q, which we view as a subfield of C, and n distinct formal power series (Puiseux expansions at infinity) vi .u/ D

1 X

˛k,i uk=i , with ˛ i ,i ¤ 0 .i D 1, : : : , n/,

(8.8)

kD i

where for each i , the following hold: i , i 2 Z, i  1, all ˛k,i ’s belong to K, the formal identity g.u, vi .u// D 0 holds, and i is minimal subject to the restriction that no proper divisor of i divides all k  i with ˛k,i ¤ 0. (b) Any formal power series v.u/ satisfying the formal identity g.u, v.u// D 0 and having properties analogous to those of the series (8.8), even without the requirement that the coefficients of v.u/ be algebraic, necessarily coincides with one of the above n series. 1 2

We will need them only at the final stage of the resolution, in Section 10.2.4. We note that the terminology “Puiseux series” is absent from these two books.

81

Section 8.2 Puiseux series

(c) The formal identity g.u, v/ D p0 .u/ holds, where p0 .u/ is the coefficient of

n Y

.v  vi .u//

iD1 n v in

g.u, v/.

(d) Each series (8.8) converges for juj > B0 , where B0 is the maximum modulus of the roots of the polynomial Resv .g, @g / 2 ZŒu. @v P k (e) For each i the function t 7! vi .t i / D 1 kD i ˛k,i t is analytic and one-to1=i

one into the punctured disk with centre at the origin and radius B0

.

Proof. We do not present here a proof in the strict sense of the word. The proofs of statements (a) through (d) are scattered in the bibliography. Statements (a), (b), and (c) are classical results about Puiseux series. They can be found in classical books such as [4, Chapter II] and [66, Chapter IV], though in slightly different form. In these books the notion of parameterisation is used in order to express the solutions .u, v/. More specifically, instead of (8.8), the parameterisation u D t i v D vi .t i / D

1 X

˛k,i t k

kD i

is used. Here we prefer to follow [67, §3]. Statement (d) seems widely known. However, we could not find an easily accessible reference where this is explicitly stated and proved. Implicitly it can be derived, for example, from [4, Chapter II] (especially §13). Statement (e) is found, for example, in [4, Theorem 13.1] and what precedes this theorem. For the injectivity proof of t 7! vi .t i / we need the somewhat technical requirement in (a) on the minimality of i . Remarks. (1) The Puiseux expansions (8.8) can be computed algorithmically by means of Newton polygons; see for instance [66, Chapter IV, §3]. An interesting refinement of this process is found in [69] with an added discussion on complexity matters; see also [68, §2] and [70, §3]. (2) The routine puiseux of MAPLE computes the Puiseux expansions of an algebraic function or, rather, the conjugacy classes of these expansions; by this we mean the following. Fix a primitive i -th root of unity . For every m 2 ¹0, 1, : : : , i  1º and every  2 Gal.K=Q/ define the formal series vi .u, m,  / D

1 X kD i

 .˛k,i / mk uk=i ;

82

Chapter 8 Linear form for the general elliptic equation

in particular, vi .u, 0, id/ D vi .u/

.i D 1, : : : , n/.

The series vi .u, m,  / also satisfies g.u, vi .u, m,  // D 0, therefore, by Fact 8.2.1(b), it must coincide with another series (8.8), say with vj .u/ D vj .u, 0, id/. We then say that vj .u/ and vi .u/ belong to the same conjugacy class. The n series (8.8) are thus partitioned into disjoint conjugacy classes (see after relation (3.5) in [67]). Therefore, the computation of a smaller set of series (8.8) composed of representatives of each conjugacy class is sufficient for the computation of all Puiseux series (8.8). ExampleStep 2 . (continued from the end of Step 1, page 80) With the aid of MAPLE we compute two conjugacy classes of Puiseux series solving g.u, v/ D 0. The first class consists of the series 1 1105 1=2 6529121 3=2 390266 2 iu 138u1 C iu C u C   , v1 .u/ D i u1=2 C  2 24 1152 9 p where i D  1, and 1 1105 1=2 6529121 3=2 390266 2 v2 .u/ D i u1=2 C C  138u1   iu iu u C , 2 24 1152 9 The second class consists of the series10 5 1 1271288 4=3 275 1=3 3317 2=3 v3 .u/ D u1=3  C  C 92u1  u u u 3 9 81 729 9364963 5=3  C  , u 6561 1 275 1=3 3317 2 2=3 1271288 4=3  C 92u1  C !u ! u !u 3 9 81 729 9364963 2 5=3  C  , ! u 6561

v4 .u/ D ! 2 u1=3 

1 1271288 2 4=3 275 2 1=3 3317 2=3  C 92u1  C ! u !u ! u 3 9 81 729 9364963 5=3  C  , !u 6561 where 10 0 ! is a primitive cubic root of unity. In the notation of Fact 8.2.1(a), K D Q.i , !/. Also, mi D 1, i D 2 for i D 1, 2 and mi D 1, i D 3 for i D 3, 4, 5. The constant B0 , defined in Fact 8.2.1(d) is approximately equal to 483.945; we round to B0 D 484. Example continued on page 86. v5 .u/ D !u1=3 

Computing the number field K . The extension K=Q mentioned in Fact 8.2.1(a), can be computed by an algorithm of J. Coates which is implicitly contained in the proof of [8, Lemma 3].

83

Section 8.2 Puiseux series

First, define the polynomials with integer coefficients pj , j D 1, : : : , n by writing g.u, v/ D

n X

pj .u/v nj .

j D0

For any i D 1, : : : , n it suffices to show that there exists an index k0,i  0 such that, ˛kC 2 Q.˛ , : : : , ˛ Ck0,i / for all k  k0,i . Therefore, let us fix i 2 ¹1, : : : , nº and subsequently, in order to simplify notation, omit the subscript i from i , ˛k,i , i and vi .u/ in (8.8). Compute the integers  D i and D i by the first few steps in the construction of the Newton polygon. Choose a non-negative integer N such that  degpj C .n  j / C N  0 .j D 0, : : : , n/ with equality for at least one subscript j , and put Pj .X / D pj .X  /  X .nj /CN and G.X , Y / D

n X

.j D 0, : : : , n/

Pj .X /  Y nj .

j D0

Then G 2 ZŒX , Y , and it is straightforward to check that G.X , y.X // D 0 identically in X , where y.X / D

1 X

˛kC X k .

(8.9)

kD0

Before giving Coates’ algorithm, let us make a notation remark: For any polynomial h over any field, in one or more variables X , : : :, when we write r D ordX .h/, we mean that X r appears in some monomial of h with non-zero coefficient, and r is minimal with respect to this property. 10 6 Computing the finite extension Q.˛ , ˛C1 , : : :/ DESCRIPTION: Compute k0  0 such that ˛ Ck 2 Q.˛ , : : : , ˛ Ck0 / INPUT: G.X , Y / as above, m D degX G, D .2n  2/m C 1. OUTPUT: k0 . INITIAL VALUES: G0 .x, y/ G.x, y/, k 0.

Hk .Y / Gk .0, Y /. if degHk D 1 then go to () else Define uk by Hk .uk / D 0 r ordX .Gk .X , uk C X Y // Gk .X , Y / X r Gk .X , uk C X Y / k kC1 goto () endif k () k0 ()

END

8k  k0 .

84

Chapter 8 Linear form for the general elliptic equation

By the proof of [8, Lemma 3], the polynomials Hk .k D 0, 1, 2, : : : / are not identically zero, their degrees are in non-increasing order and degHk0 D 1 for some k0  , so that for all k  k0 the polynomial Hk is of degree 1. Now for each k  0, the algebraic number ˛kC in (8.9) is a root of Hk , and therefore assumes one of the possible values of uk . By the linear character of Hk for all k  k0 , it follows that for k > k0 , the coefficient ˛kC is uniquely determined by the previous ˛’s and ˛kC 2 Q.˛ , ˛ C1 , : : : , ˛ Ck0 /.

8.3 Large solutions We focus our interest to “large” integer solutions .u, v/ of (8.1), since solutions with bounded juj are easily located. A first condition that we impose is that juj > B0 , where B0 is defined in Fact 8.2.1(d). Since the solutions .u, v/ of g D 0 with u < 0 are the solutions of g D 0 with u > 0, where g.u, v/ D g.u, v/, in the description of the method we can assume that u > B0 . We need first a lemma concerning real solutions .u, v/ with u > B0 . Lemma 8.3.1. Let g.u0 , v0 / D 0 with u0 , v0 2 R and u0 > B0 . Among the n series (8.8) there is exactly one vs .u/ with all its coefficients ˛k,s real algebraic numbers such that v0 D vs .u0 /, where u1=s is the real s -th root of 1=u. Q Proof. By Fact 8.2.1(c) we have 0 D g.u0 , v0 / D p0 .u0 / niD1 .v0  vi .u0 //. We assumed that u0 > B0 , therefore u0 is not a root of the polynomial Resv .g,

@g / D .1/n.n1/=2 p0 .u/  discv .g.u, v//; @v

(8.10)

in particular, p0 .u0 / ¤ 0. Consequently, v0 D vs .u0 / for some s 2 ¹1, : : : , nº. By Fact 8.2.1(a), the coefficients ˛k,s (k D s , sC1 , : : :) in (8.8) are algebraic numbers and now it suffices to show that all of them are real. We see this as follows. Let 0 2 Gal.K=Q/ be the automorphism which is obtained by restricting the complex conjugation automorphism of C to K. Then, by Remark 2 on page 81, vs .u, 0, 0 / D vj .u/ for some j . This means that the coefficients of the series vj .u/ are complexconjugates of the corresponding coefficients of vs .u/. Suppose now that some coefficients of this last series are not real. Then the series vj .u/ and vs .u/ are distinct. Consequently,Qj ¤ s and vj .u0 / D vs .u0 / D v0 D v0 D vs .u0 /, so that g.u0 , v/ D p0 .u0 / niD1 .v  vi .u0 // 2 CŒv is a non-zero polynomial divisible by .v  vs .u0 //.v  vj .u0 // D .v  v0 /2 , hence it has a root of multiplicity greater than 1. But, as already mentioned, u0 is not a root of the right-hand side of (8.10), therefore 0 ¤ discv .g.u, v// juDu0 D discv .g.u0 , v/, which means that g.u0 , v/ has no multiple root. This contradiction proves that all coefficients of the series vs .u/ are real and completes the proof of the lemma.

85

Section 8.3 Large solutions

Notations and assumptions. From now on and until the end of the chapter, E will denote the elliptic curve, a representative of which is the model C defined in (8.1), another one being the model E referred to in Proposition 8.1.1. We will denote by P the generic point of E such that the coordinates of P C D .u.P /, v.P // are integers and u.P / > B0 ; later we will require that u.P / be larger than an explicit constant larger than B0 . Then, by Lemma 8.3.1, v.P / D vs .u.P // for some s 2 ¹1, : : : , nº, where vs is as in Fact 8.2.1(a), with all its coefficients real algebraic numbers. For a positive real number u, when we write uk=s we mean the unique s -th real root of uk . Referring to the birational transformation (8.3), we have the following proposition: Proposition 8.3.2. The limit limu!C1 X .u, vs .u// exists in R [ ¹˙1º. If this limit is ¤ ˙1, then it is a real algebraic number that can be explicitly computed. Next, define x.u/ D X .u, vs .u//, y.u/ D Y.u, vs .u//. Then, there exists a B1  B0 such that, in the interval .B1 , C1/ the functions x and y are continuous, x is strictly monotonous and y does not change sign. Proof. First note that g.u, v/ cannot be a factor of either the numerator or the denominator of the rational function X .u, v/. For, otherwise, the model C could be injectively mapped into a straight line, which is impossible for a curve of genus 1. 1= Next, put u D t s with t 2 R and 0 < t < B0 s . Then, by Lemma 8.3.1 and Fact 8.2.1(a), x.u/ D X .t

s

,

1 X kD s

0

˛k,s u

k=s

00

ˇt C ˇ 0 t C ˇ 00 t C : : : D x.t /, (8.11) /D  t  C  0 t 0 C  00 t 00 C : : :

where ˇ, ˇ 0 , ˇ 00 , : : : ,  ,  0 ,  00 , : : : are non-zero real algebraic numbers and  < 0 < 00 < : : : , < 0 < 00 < : : : are rational integers. This shows that 8 ˆ if  D , , u!1 ˆ : sgn.ˇ= /1 if  < . By Proposition 8.1.1, the rational function X is explicitly computable and the same is true for the ˛k,s due to the algorithm on page 83. It follows that ˇ,  and x0s can be explicitly computed. Next, since the polynomial g does not divide neither denominator of X or Y, there exists a positive constant M1  B0 such that, if g.u, v/ D 0 and u > M1 , then .u, v/ is not a zero of neither denominator of X or Y. This implies that in the interval .M1 , C1/, the functions x and y are continuous, because X and Y are rational functions and vs is continuous (cf. Fact 8.2.1(e)).

86

Chapter 8 Linear form for the general elliptic equation

Concerning the strict monotonicity of the function x near C1, it suffices to show the strict monotonicity of the right-hand side of (8.11) near 0C . But this function in t is 0 00 of the form  t C  0 t C  00 t C    , where  ,  0 ,  00 , : : : are non-zero real numbers and   D < 0 < 00 < : : :. As t ! 0C the derivative of this function is non-zero with constant sign, and this proves our claim. 0 00 In complete analogy, we can express y as a series t C  0 t C  00 t C    , where 0 00 0 00 ,  ,  , : : : are non-zero real numbers and < < , : : :. Sufficiently close to 0C , such a series, clearly, does not change sign. Remark. In specific numerical examples the constant B1 can be explicitly calculated, as described in Section 8.5. Now we define a point P0 that will be involved in the linear form L.P / of the main result of this chapter, namely, Theorem 8.7.2. def

Definition 8.3.3. Let x0s D limu!1 X .u, vs .u// and denote by P0 D P0s the point of E with ´ .x0s , y0s / if x0s ¤ ˙1 , P0E D O if x0s D ˙1 2 D f .x / and y  0. where y0s 0s 0s

ExampleStep 3 . (continued from the end of Step 2, page 82) The only Puiseux series with all its coefficients real is v3 .u/. Therefore, according to Lemma 8.3.1, for every integer solution .u, v/ of (8.4) with juj  484 we have v D v3 .u/. Thus, in the notation of Proposition 8.3.2, x.u/ D X .u, v3 .u// and, putting u D t 3 D t 3 we write x.u/ as a series in t (cf. relation (8.11)) x.t / D 2169 C 8883t  52002t 2 C 546057t 3 C O.t 4 /. Then, according to Definition 8.3.3, P0 D P03 D .2169, 79947=2/. Example continued on page 90.

8.4 The elliptic integrals In this section we find an explicit relation between two elliptic integrals. The proposition below is analogous to Proposition 6.4 in the case of the quartic elliptic equation. We keep the notation of previous sections; in particular that in (8.2) and (8.3). Proposition 8.4.1. Let P 2 E with u.P / > B0 and v.P / D vs .u.P //.3 Let x, y and B1 be as in Proposition 8.3.2. For u > B1 we define Yu .u, v/  gv .u, v/  Yv .u, v/  gu .u, v/ G.u, v/ D 2 , 3X 2 .u, v/ C A 3

Cf. “Notations and assumptions” on page 85.

87

Section 8.4 The elliptic integrals

where subscripts u or v in functions indicate partial derivatives with respect to u or v, respectively, and g.u/ D G.u, vs .u//. Let " D sgn.y.u//, so that we have q y.u/ D " f .x.u//. Then,

Z

C1

u.P /

g.u/ du D" gv .u, vs .u//

Z

x0s

x.P /

p

dx f .x/

.

(8.12)

Proof. In this proof, if R 3 u 7! h.u/ 2 R is a differentiable function, then h0 .u/ will denote the derivative of h. p Differentiating y.u/ D Y.u, vs .u// and y.u/ D " f .x.u// we respectively get

and

y0 .u/ D Yu .u, vs .u// C Yv .u, vs .u//  vs0 .u/.

(8.13)

3 x.u/2 C A 0  x .u/. y0 .u/ D " p 2 f .x.u//

(8.14)

The relation g.u, vs .u// D 0 implies gu .u, vs .u// C gv .u, vs .u//vs0 .u/ D 0. Solving for vs0 .u/ and substituting in the relation which results on equating the right-hand sides of (8.13) and (8.14), gives p f .x.u// 0 . (8.15) x .u/ D " g.u/  gv .u, vs .u// In view of the strict monotonicity of the function x we can make the change of variable in the integral below Z Z C1 Z x0s dx xDx.u/ C1 x0 .u/ du .8.15/ g.u/ du p D p D " , u.P / u.P / gv .u, vs .u// x.P / f .x/ f .x.u// as claimed. For Ellog it is important that the integrand in the left-hand side of (8.12) tends to zero as u ! C1. This is made precise in the proposition below. Proposition 8.4.2. There exists a sufficiently large constant B2  B1 such that, if u  B2 , then ˇ ˇ ˇ ˇ g.u/ 1 ˇ ˇ , (8.16) ˇ g .u, v .u// ˇ  c9 juj v s where   s1 and c9 is a positive constant independent from u.

88

Chapter 8 Linear form for the general elliptic equation

Proof. We write (8.15) as an equation of differentials g.u/ dx D du. y gv .u, vs .u// The equation y2 D f .x/ is parameterised by x D x.t / D t 2 , Z

hence,

y D y.t / D t 3 C 12 At C 12 Bt 3  18 A2 t 5  14 ABt 7 C O.t 9 /, Z

x0 .t / dt D 2 C At 4 C Bt 6 C O.t 8 / y.t / R and we see that there are no singularities in the expansion of dx=y. This, by definition, means that the elliptic integral is of the first kind (see [4, §24]). Therefore, the R same is true for the integral .g.u/=gv .u, v// du, implying that, for any parameterisation .u, v/ D .u.t /, v.t // of g.u, v/ D 0, the t -expansion of dx D y

du g.u/  gv .u.t /, v.t // dt has no negative powers of t . Applying this conclusion to the parameterisation u.t / D t s ,

v.t / D vs .t s / D

1 X

˛k,s t k ,

kD s

and taking into account that du=dt D s t s 1 , we are led to the inequality ord t

g.t s /  s C 1.4 gv .t s , vs .t s //

This means that gv

g.t s / D t 1Cs C .ˇ0 C ˇ1 t C ˇ2 t 2 C    /, s // .t s

.t s , v

  0, ˇ0 ¤ 0.

Putting t s D u in the relation above we conclude that g.u/ D u1 .ˇ0 C ˇ1 u1=s C ˇ2 u2=s C    /, gv .u, vs .u//

 D .1 C /=s  1=s .

For sufficiently large u, the absolute value of the right-hand side is bounded above by c9 juj1 , as claimed. Remark. In specific numerical examples the constants B2 , c9 and  of Proposition 8.4.2 can be made explicit according to Section 8.6. 4

The function ord has been defined just before the algorithm on page 83.

Section 8.5 Computing in practice B1 of Proposition 8.3.2

8.5

89

Computing in practice B1 of Proposition 8.3.2

In Proposition 8.3.2 we introduced the constant B1 , the existence of which is guaranteed by a non-constructive proof. In any specific equation, however, we must explicitly compute such a B1 , and in this section we show how we can do this. First we state a simple lemma. Lemma 8.5.1. Let F be a polynomial in two variables with real coefficients and let V : R ! R be a continuous function, such that F .u, V .u// D 0 for juj > U0 , where U0 is a positive constant. Let R be the set of all real roots of the polynomial F .0, Y /, and S D ¹u : juj > U0 & V .u/ 2 Rº. Put .Umin , Umax / D .U0 , U0 / or .Umin , Umax / D .min S, max S/ according to whether S is or is not empty. Then the function V keeps a constant sign in the interval .Umax , C1/ and does so in the interval .1 , Umin /. Proof. Suppose that u2 > u1 > Umax and V .u2 /V .u1 / < 0. Then, by the continuity of V , there exists a u0 2 .u1 , u2 /, such that V .u0 / D 0. This means that u0 2 S , hence u0  Umax . But u0 > u1 > Umax , a contradiction. Similarly we arrive at a contradiction if we assume that u2 < u1 < Umin and V .u2 /V .u1 / < 0. We will use this lemma in order to show how to compute M1,min and M1,max absolutely larger than B0 , such that y keeps a constant sign in the interval .M1,max , C1/ and does so in the interval .1 , M1,min /. Let D.u, v/ be the square-free part of the product of the denominators of X and Y. Since g is an absolutely irreducible polynomial and does not divide neither denominator of X or Y, the resultant Resv .D, g/ is a non-zero polynomial in u; we denote by R the maximum absolute value of its real roots. Let any u with juj > max¹B0 , Rº. After clearing out the denominator in y.u/ D Y.u, vs .u//, we obtain the polynomial relation H1 .u, vs .u/, y.u// D 0. On the other hand, we also have g.u, vs .u// D 0 and we can eliminate vs .u/ from the last two relations, using resultants. We thus obtain a polynomial relation R1 .u, y.u// D 0. Now, applying Lemma 8.5.1 with F D R1 and V D y we compute values Umin , Umax (in the notation of the lemma) and we take M1,max D dmax¹Umax , B0 , Rºe and M1,min D bmin¹Umin , B0 , Rºc. Next we show how to compute M2,mi n and M2,max such that in both intervals, .M2,max , C1/ and .1 , M2,mi n /, x is strictly monotonous. Again, let any u with juj > B0 . Clearing out the denominator in x.u/ D X .u, vs .u// gives a polynomial relation H2 .u, vs .u/, y.u// D 0. From this relation and g.u, vs .u// D 0 we eliminate vs .u/, using resultants. Thus we obtain a polynomial relation R2 .u, x.u// D 0. Differentiating with respect to u gives @R2 @R2 .R2 .u, x.u// C .R2 .u, x.u//  x0 .u/ D 0, @u @x

(8.17)

90

Chapter 8 Linear form for the general elliptic equation

where x0 means derivative with respect to u. This is a polynomial relation H3 .u, x.u/, x0 .u// D 0. Eliminating x.u/ from the last relation and R2 .u, x.u// D 0, we get a polynomial relation R3 .u, x0 .u// D 0 and we apply Lemma 8.5.1 with F D R3 and V D x0 to obtain the values Umin and Umax (in the notation of the lemma). Then, it suffices to take M2,max D dmax¹Umax , B0 , Rºe and M2,min D bmin¹Umin , B0 , Rºc. Finally, B1 D max ¹jMi,min j , jMi,max jº. 1i2

ExampleStep 4 . (continued from the end of Step 3, page 86) The constant B0 appearing in Fact 8.2.1(d) is already computed: B0 D 484. Therefore we will search for integer solutions .u, v/ with juj  484. By (8.6), R D 1, so that the requirement u > R is already fulfilled. An easy search shows that the only integer solutions with juj  483 are .u, v/ D .0, 0/, .1, 3/, .243, 3/. We first calculate M1,max and M1,min . In the notation of our previous discussion, we calculate H1 .u, vs .u/, y.u//, a straightforward task. It turns out that H1 is the sum of 19 monomials and degu H1 D 4, degvs H1 D 4, degy H1 D 1. Then, using MAPLE, the computation of R1 .u, y.u// presents no difficulty; the result is the product of 81u5 .u  1/12 with a polynomial in u and y, let us denote it by R10 , which is the sum of 24 monomials with some coefficients having 36 decimal digits, degu .R10 / D 3 and degy D 5. Since we have already assumed that juj > 484, we must have R10 D 0 and we apply Lemma 8.5.1 with F D R10 , U0 D B0 D 484, V D y. We have F .0, y/ D .2y C 81029/.2y  35626473/4 , therefore R D ¹ 1 , 2 º, where 1 D 81029=2 and 2 D and 2 D 35626473=2. The solutions of the equation y.u/ D 1 are approximately 2339.17722, 346.44760 and 0. Hence, in the notation of Lemma 8.5.1, S D ¹2339.17722º, U1,min D U1,max D 2339.17722 and, consequently, M1,max D 484 and M1,min D 2340, which implies that y does not change sign in the interval .1 , 2340/, and the same is true for y in the interval .484 , C1/. Next we compute M2,max and M2,min . The polynomial H2 .u, vs .u/, x.u// has 13 monomial terms and degu H2 D 3, degvs H2 D 4, degx H2 D 1. The resultant R2 .u, x.u// is the product of 81u5 .u  1/8 with a polynomial R20 .u, x/ which is the sum of 18 monomials of degree 2 with respect to u and degree 5 with respect to x; its coefficients have at most 23 decimal digits; then, necessarily, R20 D 0. According to our discussion before the example, the left-hand side of (8.17) is a polynomial H3 in u, x and x0 , linear in x0 . Eliminating x from R20 D 0 and H30 D 0, we obtain R3 .u, x0 .u// D 0, where R3 .u, x0 .u// D Resx .R20 , H3 /. It turns out (with the aid of MAPLE) that R3 is the product of a 22 decimal-digit integer with the square of a cubic polynomial in u (only), the roots of which belong to the interval .540 , 311/, and a polynomial R30 in u and x0 which is the sum of 35 monomials, having coeffi-

91

Section 8.6 Computing in practice B2 and c9 of Proposition 8.4.2

cients with at most 33 decimal digits; also, degu .R30 / D 10 and degx0 .R30 / D 5. If we assume that juj  540, then, necessarily, R30 D 0 and we apply Lemma 8.5.1 with F D R30 , V D x0 , U0 D B0 D 484. Now F .0, x0 / D .271x0  299/ times a large constant, therefore R D ¹299=271º. The approximate solutions of R30 .x, 299=271/ D 0 are 932.85436, 738.24160, 173.11938 and 0, therefore, S D ¹932.85436 : : : , 738.24160 : : :º. Consequently, M2,mi n D 933, M2,max D 484 and Lemma 8.5.1 implies that x0 is non-zero in the interval .1 , 933/ and has a constant sign, hence x is strictly monotonous. Similarly, x is strictly monotonous in the interval .484 , C1/. Finally, by the displayed formula for B1 we obtain B1 D 2340. Therefore, from now on, we will assume that juj > B1 D 2340. Example continued on page 91.

8.6

Computing in practice B2 and c9 of Proposition 8.4.2

At first we work as in Section 8.5. We denote by I D I.u/ the left-hand side of (8.16) and clear out the denominator to obtain a polynomial relation H4 .u, vs .u/, I/ D 0. Using resultants we eliminate vs .u/ from the last relation and g.u, vs .u// D 0. Thus we obtain a polynomial relation I m C q1 .u/ I m1 C    C qm1 .u/ I C qm .u/ D 0,

(8.18)

where qi .u/ 2 Q.u/ for i D 1, : : : , m. We distinguish the cases of positive and negative I. Let I be positive. For juj  B2 , where B2  B1 is sufficiently large (how large can be made explicit in each specific numerical example), each qi .u/ has constant sign. Let qj .u/, qj 0 .u/, : : : with 1  j < j 0 < : : :  m be the strictly negative coefficients in (8.18) and let k be their number. By the so-called Cauchy rule,5 we have 0 I  max¹.kjqj .u/j/1=j , .kjqj 0 .u/j/1=j , : : :º, from which we find an inequality of the form I  constant  u1 , where the “constant” and  > 0 is explicit. Next, we replace I by I and repeat the argument to obtain a similar upper bound for I. Combining the two upper bounds we get an upper bound for jIj of the form c9  u1 , where c9 is an explicit positive constant. ExampleStep 5 . (continued from the end of Step 4, page 91) Following the instructions above we compute H4 .u, vs .u/, I/ D u2 .u  1/4 H40 .u, vs .u/, I/, 5

By this we mean the following: Let f .X / D X n C a1 X n1 C    C an 2 RŒX  and let al , am , : : : be the negative coefficients, their total number being k, where l > m > : : :. Then every real root of f .X / satisfies  max¹.kjal j/1= l , .kjam j/1=m , : : :º; the result and its naming is found in [31, Chapter V.4]; probably, to other people’s minds, “Cauchy’s rule” (for the roots of a polynomial) means a result of this type, but not quite the same.

92

Chapter 8 Linear form for the general elliptic equation

where H40 .u, vs .u/, I/ is the sum of 84 monomials, degu .H40 / D 7, degvs .H40 / D 12 and, of course, H40 is linear in I. Then, Resvs .H40 .u, vs .u/, I/, g.u, vs .u// D constant  u12 .u  1/16 .81000u C 19902511/.784u  17846163/h.u/R4 .u, I/, where h.u/ is a second degree polynomial. In the right-hand side, the roots of the non-constant factors, except for R4 .u, I/ belong to the interval .696 , 22763/. Also, R4 .u, I/ is the sum of 12 monomials, degu .R4 / D 7 and degI .R4 / D 5. If we assume that juj  22763, then R4 .u, I/ D 0 and we write the last relation as follows: I 5 C q2 .u/ I 3 C q3 .u/ I 2 C q4 .u/ I C q5 .u/ D 0,

(8.19)

(q1 .u/ D 0) with 4 2025u2 C 1491042u C 318440176  9 u q.u/ 32 3u C 1355 q3 .u/ D  9 u q.u/ 1175056 q4 .u/ D 243 u3 q.u/ 32 q5 .u/ D , 243 u3 q.u/ q2 .u/ D

where q.u/ D26244u4 C 20515275u3 C 1099852416u2  2071134904704u  374185039449856. Now we work as follows. Consider q2 .u/. Its numerator has no real roots and those of the denominator belong to the interval .484 , 318/. But we have already assumed that juj  22763 and we compute that q2 .22763/ < 0 and q2 .22763/ > 0. Hence, if u  22763, then q2 .u/ < 0 and if u  22763, then q2 .u/ > 0. In an analogous way we check that, if juj  22763, then q3 .u/ > 0, and if u  22763, then both q4 .u/ and q5 .u/ are negative, and if u  22763, then both q4 .u/ and q5 .u/ are positive. In view of the above discussion, if u  22763, then qi .u/ > 0 for i D 2, 3, 4, 5 and, consequently, in (8.19), I < 0. Setting I D J < 0 we obtain the equation J 5 C q2 .u/ J 3  q3 .u/ J 2 C q4 .u/ J  q5 .u/ D 0,

(8.20)

where now the strictly negative coefficients are q3 .u/ and q5 .u/. By Cauchy’s rule, 0 < J < max¹.2q3 .u//1=3 , .2q5 .u//1=5 º. We have q3 .u/ D

32 3u4 C 1355u3 4   u < .2=27/3  u4 9 q.u/

Section 8.6 Computing in practice B2 and c9 of Proposition 8.4.2

and q5 .u/ D

93

32 u4   u7 < .23 =313 /  u7 . 243 q.u/

Hence, by Cauchy’s rule, 0 < I D J < max¹.24=3 =27/ u4=3 , .24=5 =313=5 / u7=5 º < 0.1001 u4=3 Consider now (8.19) when u < 22763. The strictly negative coefficients in (8.19) are q2 .u/, q4 .u/ and q5 .u/. Let first I  0. Then, by Cauchy’s rule, 0 < I < max¹.3jq2 .u/j/1=2 , .3jq4 .u/j/1=4 , .3jq5 .u/j/1=5 º. We have jq2 .u/j D

4 j2025u4 C 1491042u3 C 318440176u2 j   juj3 9 jq.u/j

jq4 .u/j D

1175056 u4   juj7 243 jq.u/j

jq5 .u/j D

32u4  juj7 . 243 jq.u/j

From these relations we easily obtain the following upper bound for I: 0 < I < 0.11 juj7=5 . Next, let I < 0. We put I D J with J > 0, so that the equation (8.20) holds. Since u  22763, the strictly negative coefficients in (8.20) are now qi .u/ with i D 2, 3, 4. By Cauchy’s rule, 0 < J < max¹.3jq2 .u/j/1=2 , .3jq3 .u/j/1=3 , .3jq4 .u/j/1=4 º. Working as we did a few lines above, we obtain 0 < I D J < 0.11juj4=3 . Combining all our partial results above we come to the following conclusion: If juj  22763, then the left-hand side of (8.16) is bounded above by 0.11 juj4=3 , hence, in the notation of Proposition 8.4.2, B2 D 22763,  D 1=3 D 1=s and c9 D 0.11. Remark. An easy search for integer solutions of g.u, v/ D 0 in the range juj < 22763 reveals no further solutions than those already found, namely, .u, v/ D .0, 0/, .1, 3/ and .243, 3/. Example continued on page 96.

94

Chapter 8 Linear form for the general elliptic equation

8.7 The linear form L.P/ and its upper bound The goal of this section is the definition of a linear form L.P / in elliptic logarithms and the proof of Theorem 8.7.2 which provides us with an explicit upper bound for jL.P /j. We keep the “Notations and assumptions” on page 85, except that now the condition u.P / > B0 is replaced by the stronger condition u.P / > B2 (see the conclusion in page 93). Note that the case u.P / < B2 is treated in a completely analogous manner, as the solutions of g.u, v/ D 0 with negative u coincide with the solutions of g.u, v/ D 0 with positive u, where g.u, v/ D g.u, v/. We also keep the notations, assumptions and results of Chapter 4. In particular, e1 denotes the largest real root of the polynomial f .X / in (8.2) if this polynomial has three real roots and e1 is the only real root of f .X / otherwise. We distinguish two cases regarding the relative position of e1 and x0s (cf. Definition 8.3.3 and Proposition 8.3.2). We have P C D .u.P /, vs .u.P /// and P E D .x.P /, y.P // D .x.u.P //, y.u.P ///; see Proposition 8.3.2. By that proposition, the continuous function u 7! x.u/ is strictly monotonous in the interval .B1 , C1/ with limu!C1 x.u/ D x0s . A moment’s thought then shows that, if x0s  e1 then x.P / D x.u.P //  e1 , and if (in the case that also e2 and e3 are real) e3  x0s  e2 , then e3  x.P / D x.u.P //  e2 . In other words, P E is a point of E0 .R/ or E1 .R/ (cf. beginning of Section 3.5), according to whether P0E is a point of E0 .R/ or E1 .R/, respectively. By Proposition 8.3.2 we also know that y.u/ has a constant sign when u 2 .B1 , C1/, which we will denote by ". By (8.12) we have Z Z x0s C1 g.u/ du dx (8.21) D " p u.P / gv .u, vs .u// x.P / f .x/ and we distinguish two cases. (1) P0 2 E0 .R/. Then P E 2 E0 .R/ and using (3.32),6 we write the right-hand side of (8.21) as follows: Z C1 Z C1 Z x0s dx dx dx D  D 2"P l.P /  2 l.P0 /. p p p x.P / x0s x.P / f .x/ f .x/ f .x/ (2) P0 2 E1 .R/.7 Then, P E 2 E1 .R/. Making use of the point Q2 , that point of E with Q2E D .e2 , 0/ (cf. Lemma 6.6), we write the right-hand side of (8.21) as follows: Z e2 Z e2 Z x0s dx dx dx D  p p p x.P / x.P / x0s f .x/ f .x/ f .x/ 6 7

The fact that in the definition of P0 we assumed y.P0 /  0 is also used. Case possible only if all three roots of f .X / are real.

95

Section 8.7 The linear form L.P / and its upper bound

Z D

C1

x.P CQ2 /

dx

Z

 p f .x/

C1

x.Q2 CP0 /

p

dx f .x/

D 2"P l.P /  2 l.P0 /.

(by Lemma 6.6) (by (3.32))

Thus, in both cases (1) and (2) we have, in view of (8.21), ˇZ ˇ 1 ˇ C1 g.u/ du ˇˇ jl.P / C "P l.P0 /j D ˇˇ . 2 u.P / gv .u, vs .u// ˇ

(8.22)

Now we will prove a proposition which will permit us to compute an upper bound for the right-hand side of (8.22) in terms of u.P /. Proposition 8.7.1. Let us write the relation g.u, v/ D 0 in the form v n C a1 .u/ v n1 C    C an1 .u/ v C an .u/ D 0

(8.23)

where the ai ’s are polynomials in u. Let B3 be a constant larger than every root of every non-zero polynomial ai . If P C 2 C.Z/ and u.P / > max¹B2 , B3 º, then h.x.P //  c10 C c11 log ju.P /j,

(8.24)

where, h./ denotes absolute logarithmic height and c10 , c11 are explicitly computable positive constants, independent from P . Proof. We have v.P / D vs .u.P //. We write X D F1 =F2 for some relatively prime polynomials with rational integer coefficients. For simplicity in the notation of this proof let us put u.P / D u and v.P / D v, so that h.x.P // D h.X .u.P /, v.P //  log max¹jF1 .u, v/j, jF2 .u, v/jº D log jFj .u, v/j for the proper choice of j D 1, 2. Let 1  i  n be such that ai is a non-zero polynomial. Then ai .u/ does not change sign as u runs through the values > B2 . Therefore, for u > B2 , it makes sense to distinguish all the (strictly) negative coefficients, say am1 .u/, am2 .u/, : : : , amk .u/, in the left-hand side of (8.23). If v  0, Cauchy’s rule8 implies that 0  v  max .kjami j/1=mi 1ik

0

and, clearly, the right-hand side is bounded by ˛ 0 uˇ , where ˛ 0 and ˇ 0 are explicit positive constants (ˇ 0 2 Q). If v < 0, we put v D w with w > 0 and we rewrite (8.23) as w n C b1 .u/ w n1 C    C bn1 .u/ w C bn .u/ D 0, where bi D ˙ai for 00 i D 1, : : : , n. As before, Cauchy’s rule implies a relation of the form 0 < w  ˛ 00 uˇ and, combining the two upper bounds we obtain an explicit bound jvj  ˛uˇ . 8

See Footnote 5.

96

Chapter 8 Linear form for the general elliptic equation

P Next, we write Fj .u, v/ D .k,l/ ak,l uk v l and let c11 D max.k,l/ ¹k C lˇº, where the maximum P runs over all pairs .k, l/ for which ak,l ¤ 0. Then, jFj .u, v/j  c10 uc11 , where C10 D .k,l/ jak,l j˛ l and (8.24) holds with c10 D log C10 . ExampleStep 6 . (continued from the end of Step 5, page 93) u and a5 D u2 , hence we can take We have a1 D 0, a2 D u, a3 D 0, a4 D  271 3 B3 D 1. We put .u.P /, v.P // D .u, v/ and, according to Proposition 8.7.1, we assume that juj  max¹B2 , B3 º D B2 D 22730. Assume u > 22730. 

If v  0, then Cauchy’s rule implies 0  v  max¹.2ja4 j/1=4 , .2ja5 j/1=5 º D p 542 1=4 p 5 5 max¹. 3 u/ , 2 u2=5 º D 2 u2=5 .

If v < 0, we put v D w with w > 0 so that g.u, v/ D 0 is written as w 5 C uw 3  271 uw  u2 D 0. In the notation of our previous discussion, b1 D 0, b2 D u, b3 D 3 0, b4 D  271 u, b5 D u2 , hence 0 < v D w  . 271 u/1=4 , by Cauchy’s rule. 3 3 p Combining the two upper bounds we conclude that jvj  5 2 u2=5 . Next, assume u  22730. Then, we consider g.u, p v/ D g.u, v/ instead of 1=2 g.u, v/. Working as above we obtain the bound jvj  p 2 juj . 1=2 Thus, in general, for juj  22730 we have jvj  2 juj and, consequently, the absolute value of the numerator of X .u, v/ is, easily, bounded by p p 2710juj C 66879juj5=2 2 C 43821juj3=2 2 C 52542juj2 C 29331juj3 < 30000juj3 

and, clearly, 30000juj3 is an upper bound for the absolute value of the denominator u.u  1/2 of X .u, v/. Thus, h.x.P // D log jx.P /j D log max¹numer.jX .u.P /, v.P //j/ , denom.jX .u.P /, v.P //j/º  log.30000juj3 / D log.30000/ C 3 log ju.P /j and, consequently, c10 D log.30000/ and c11 D 3. Example continued on page 118. Theorem 8.7.2. Let E be the elliptic curve represented by the model C in (8.1), subject to the conditions at the beginning of this chapter. def Let E : y 2 D x 3 C Ax C B D f .x/ be the short Weierstrass model with A, B defined in Proposition 8.1.1 and consider that everything in Chapter 4, in particular, equation (4.1) and relations (4.4)–(4.8), refer to this particular model E. Let X be the rational function defined in Section 8.1 and let P0 be the point of E in Definition 8.3.3.

Section 8.7 The linear form L.P / and its upper bound

97

Assume that P 2 E is such that P C D .u.P /, v.P // has integer coordinates with u.P /  max¹B2 , B3 º 9 and v.P / D vs .u.P // for some Puiseux series vs (see Fact 8.2.1 and (8.8)) all the coefficients of which are real (see “Notations and assumptions” on page 85). Write P E as in (4.2), consider l.P / as in (4.6) and define L.P / D l.P / C "P l.P0 /, where "P D 1 or 1, according to whether y.P / > 0 or y.P /  0, respectively. Then   .2 C c10 / 2  2 c9  M , exp (8.25) jL.P /j  1C c11 c11 where c9 and  are as in Proposition 8.4.2,10 c10 and c11 are as in Proposition 8.7.1 and  and are defined in Propositions 2.6.3 and 2.6.2, respectively. Proof. By the definition of L.P / and relations (8.22) and (8.16) we see that jL.P /j  c9 ju.P /j . By (8.24), ju.P /j1  exp¹.c10  h.x.P ///=c11 º and, therefore, 1C   c9 .c10  h.P // jL.P /j  exp . 1C c11

(8.26)

O /  2  2 M 2 , where the right-most By Proposition 2.6.3, h.x.P //  2  2h.P inequality holds because of Proposition 2.6.2 and (4.7). Replacing for h.P / in (8.26) completes the proof.

9

For B2 see Proposition 8.4.2 and Section 8.6; for B3 see Proposition 8.7.1. also Section 8.6.

10 See

Chapter 9

Bound for the coefficients of the linear form

In each of Chapters 5, 6, 7 and 8 we defined a convenient linear form L.P /, where P denotes the point which “covers”1 the generic (“sufficiently large”) solution of the Diophantine problem considered in the chapter. Theorems 5.2, 6.8, 7.1 and 8.7.2 of the corresponding chapters provide us with an upper bound for jL.P /j as a function of M , the maximum absolute value of the coefficients of L.P /. In the present chapter we will give a lower bound for jL.P /j, again as a function of M . As it will turn out, comparing the upper and lower bound will result to an upper bound for M and, consequently, to the effective solution of the Diophantine problem under consideration; see Chapter 4 immediately after (4.8).

9.1 Lower bound for linear forms in elliptic logarithms In each of the Theorems 5.2, 6.8, 7.1 and 8.7.2, the linear form L.P / to which the corresponding theorem refers, has a direct simple relation with the linear form s l.P / D .m0 C /!1 C m1 l.P1 / C    C mr l.Pr /, t defined in (4.6). For the above displayed linear form the relations (4.3)–(4.8) are valid; in particular M D max1ir jmi j and M0 D max0ir jmi j. We put `i D l.Pi / for i D 1, : : : , r. Possibly in Theorem 6.8 and, anyway, in Theorems 7.1 and 8.7.2, another point P0 is involved. The coordinates of P0E belong to a number field which is of degree at most two in the case of Theorem 6.8 and four in the case of Theorem 7.1; for the case of Theorem 8.7.2 there is no a priori bound for the degree of the coordinates of P0E . We will also put `0 D l.P0 /. Finally, we will put m00 D m0 , except in that sub-case of Theorem 6.8 when x0 D e1 , in which we will put m00 D m0 C 2. Having agreed to the above, we can give a first uniform shape to L.P /, namely  s (9.1) !1 C m1 `1 C    mr `r C "`0 , L.P / D m00 C t

1

For this terminology, see a few lines below (4.2).

Section 9.1 Lower bound for linear forms in elliptic logarithms

8 ˆ 0 ˆ ˆ ˆ ˆ ˆ 0,

j Q j  1,

1 j< Q j  . 2

Proof. First a remark concerning notation: For any matrix M D . and any non-real 2 C we write, by definition, def

M D

a c

b d

/ 2 GL2 .Z/

a C b . c C d

Now we proceed to the proof. Let .!1 , !20 / be a fundamental pair of periods for } obtained by Theorem 3.4.1, where, in the notation of that theorem, !20 D !2 in the case of positive discriminant, and !20 D .!1 C !2 /=2 in the opposite case. Let D !20 =!1 (note that = > 0). By [45, Proposition 12.1], there exists   a0 b0 M0 D 2 SL2 .Z/ c0 d0 such that M0 satisfies both j exp  c13 .log N C c14 /.log log N C c15 /kC2 ,

where 6kC12

c13 D 2.9  10

D

2kC4 2.kC1/2

c14 D 1 C log D ,

4

2k 2 C13kC23.3

.k C 2/

k Y

(9.7)

Hi ,

iD0

c15 D 1 C hE C log D. Proof. In the case of the first “bullet”, page 99, the relation L.P / D 0 means that r0 m1 l.P1 / C    C r0 mr l.Pr / C n0 !1 D 0. By Theorem 3.5.2 (c) it follows then that m1 D    D mr D 0, hence P is a torsion point. In the case of the second “bullet”, page 99, the relation L.P / D 0 is impossible. Indeed, in this case, L.P / D 0 means that r0 m1 l.P1 / C    C r0 mr l.Pr / ˙ r0 l.P0 / C n0 !1 D 0 and the points P1E , : : : , PrE , P0E are Z-linearly independent. Since the coefficient of l.P0 / is non-zero, this contradicts Theorem 3.5.2 (c). In the case of the third “bullet”, page 100, L.P / D 0 means (cf. (9.4)) that r0 .d m1 C d1 /`1 C    C r0 .d mr C dr /`r C n0 !1 D 0 and Theorem 3.5.2 (c) implies that mi D di =d for i D 1, : : : , r. In particular, d is a common divisor of d1 , : : : , dr and, since gcd.d , d1 , : : : , dr / D 1, we conclude that d D 1 and mi D di for i D 1, : : : , r. If L.P / ¤ 0, then the conclusion of the theorem results from an almost straightforward, though careful (see remarks below), application of [12, Théorème 2.1]. Remarks on the application of S. David’s theorem. (1) In S. David’s theorem [12, Théorème 2.1] the k elliptic logarithms appearing in the linear form may come from different elliptic curves defined over a number field K. In our case, all elliptic curves coincide with the elliptic curve (4.1) defined over Q. We remind, however, that some `i ’s may be elliptic logarithms of points with non-rational coordinates, as noted on page 101. (2) For the parameter E appearing in [12, Théorème 2.1] we chose E D e. (3) A “detail” concerning canonical heights. According to our discussion in Chapter 2, the canonical height in [12, Théorème 2.1], which on page 20 ff. we denoted by hOD , is, by Proposition 2.6.1, three times the canonical height defined by J. Silverman in [45, section VIII.9], i.e. the canonical height that we adopt in this book. This O i / is multiplied by a explains why in the definition of Hi the canonical height h.P factor 3, which does not appear in S. David’s theorem.

Section 9.2 Computational remarks

105

Now we are ready state the theorem from which an explicit upper bound of M is obtained. Theorem 9.1.3. Let L.P / be the linear form appearing in any of Theorems 5.2, 6.8, 7.1 or 8.7.2. Write L.P / in the form (9.2) following the instructions in the appropriate “bullet” (page 99 ff.) and consider the constants ˛ and ˇ (cf. (9.6)), as well as the constants c12 , c13 , c14 , c15 resulting from Theorem 9.1.2. If L.P / ¤ 0, then, either M  c12 , or M 2  c18 c13 .log.˛M Cˇ/Cc14 /.log log.˛M Cˇ/Cc15 /kC2 C Cc18 log c16 Cc17 , (9.8) where, 8 ˆ in case of Theorem 5.2 .23=2 , 12 log ı, 1/ ˆ ˆ p ˆ ˆ c12 , the left inequality (9.6) implies N > c12 and then Theorem 9.1.2 implies the lower bound (9.7) for jL.P /j. In view of the right inequality (9.6), this lower bound is also valid after replacing N by ˛M C ˇ. Then the resulting lower bound, combined in a straightforward manner with the upper bound for jL.P /j already obtained from Theorem 5.2 or 6.8 or 7.1 or 8.7.2, depending on the case, leads to the inequality (9.8). Concluding remark. In (9.8), the left-hand side tends to infinity faster than the righthand side. Therefore, if M is sufficiently large, then this relation cannot hold. Consequently, if all constants in (9.8) are explicitly known, then an upper bound for M can be explicitly calculated; cf. Chapter 4, immediately after (4.8).

9.2

Computational remarks

In sections 9.3, 9.4, 9.5 and 9.6 we show by concrete examples how to apply Theorem 9.1.3 and the Concluding remark following it. In the present section we give some practical information concerning our concrete numerical computations. We make a combined use of MAGMA, PARI and a few rather simple codes based on MAPLE written by the author. In all examples we started doing all calculations with a precision of 100 decimal digits. With the exception of Section 9.4, in all other sections, the numbers resulting from the calculations are of a size much less than 10100 , so that

106

Chapter 9 Bound for the coefficients of the linear form

we consider them reliable. In Section 9.4, however, the value of the constant c13 turned out to be considerably larger that 10100 and we repeated all the computations using a 200 decimal digits precision. All packages nowadays work with this and much larger precision without the slightest problem. Anyway, in the text we write the results of our computations correct to 16 decimal digits. Computations related to elliptic curves are done using MAGMA and/or PARI. Given the vector Œa1 , a2 , a3 , a4 , a6  of the coefficients of an elliptic curve (see [45, Chapter III.1]), the routines EllipticCurve of MAGMA and ellinit of PARI create an object e containing all necessary information to work with. Minimal models are calculated by any of the routines ellminimalmodel of PARI or MinimalModel of MAGMA. For the computation, even with a very high precision, of a fundamental pair of periods for our elliptic curves (or, rather, for the }-functions associated to them) we can easily write a simple code according to the instructions in Section 3.4.4. If one does not feel comfortably with programming, any of the commands Periods.e/ of MAGMA or e.omega of PARI does the job. Elliptic logarithms can be calculated by the algorithm on page 53, for which it is not difficult to write a code. Alternatively, the routines ellpointtoz of PARI or EllipticLogarithm of MAGMA do the same thing. We remind the reader that, if for some point P we have P E 2 E1 .Q/, then l.P / is the elliptic logarithm of P C Q2 (where Q2E D .e2 , 0/) rather than the elliptic logarithm of P , which is not a real number; cf. Conclusions and remarks (1) on page 51. Probably, the point P E C Q2E has non-rational coordinates; in this case, its coordinates belong to a quadratic or a cubic number field. But computing the elliptic logarithm of such a point is not really a problem, because the algorithm on page 53 works perfectly with real numbers. Also, the routine ellpointtoz of PARI accepts input points on the elliptic curve with coordinates of type “real”, but the routine EllipticLogarithm of MAGMA would complain. In a symbolic calculation package like, for example, MAPLE, we can write a program for calculating with points on an elliptic curve over a number field, by implementing the Group law algorithm 2.3 of [45, Chapter III.2].9 The computation of the canonical height is quite a sophisticated task; see, for example, [42, 43, 75, 76].10 For the needs of our examples we used either of the routines CanonicalHeight of MAGMA or ellheight of PARI. Warning! The definition of the canonical height given in Section 2.6 agrees with the one given by Silverman [45, Chapter VIII]. The canonical height computed by the above routines of PARI and MAGMA is twice as large, because the factor 1=2 in (2.36) is missing from the definition of the canonical height adopted by PARI and MAGMA. Analogously, for the height pairing matrix, either of the routines HeightPairingMatrix or ellheightmatrix of MAGMA or PARI, respectively, is used. Note that the eigenvalues 9

This is what the author has done. relevant bibliography is far from being exhausted by these references!

10 The

107

Section 9.3 Weierstrass equation example

of the height pairing matrix are not affected by the normalisation for the canonical height that one may adopt. Warning! The routines ellheight and ellheightmatrix of PARI demand that their input data come from a minimal Weierstrass model. The extra point P0 that makes its appearance possibly in Theorem 6.8 and, anyway, in Theorems 7.1 and 8.7.2, has probably irrational coordinates and, in this case, neither PARI nor MAGMA know how to compute its canonical height. Should we need to know O 0 /, we have to implement for ourselves the algorithms of [43]. the precise value of h.P But for the application of Theorem 9.1.2 we only need a reasonably good upper bound O 0 /, which we obtain from Proposition 2.6.4. of h.P Computing the constants c12 , : : : , c18 is a straightforward but quite boring task. Therefore, we preferred to write a rather simple MAPLE code for their computation. A considerably more sophisticated code running, for example, under MAGMA, which would receive much less data – only, say, the coefficients of C and E – and return the numerical values of above ci ’s automatically, is possible. We did not attempt to write such a code as we prefer a more transparent process which is more easily under our control.

9.3

Weierstrass equation example

We use the notation, results etc. of Chapter 5. In (5.1) let us take C : g.u, v/ D 0,

g.u, v/ D v 2 C uv C v  u3  u2 C 71u C 196.

Taking in (5.2) . , , , / D .1, 5=12, 1=2, 7=24/, we obtain the model

3409 143623 x D f .x/. 48 864 The roots of f .X / are e3 D 41=6 < e2 D 31=12 < e1 D 113=12. Note also that, in Proposition 5.1, ı D 12. E : y2 D x3 

We will work with the elliptic curve E, two models of which are C and E, searching for all points P on E such that P C 2 C.Z/. In order to apply Theorem 5.2, we need that u.P / > 18. Also, we note that there are no points .u, v/ 2 C.R/ with u < 7.11, therefore we first compute all P with integer u.P / between 7 and 18; these are exactly the following: .u.P /, v.P // D.7, 1/, .7, 7/, .6, 5/, .6, 10/, .4, 5/, .4, 8/, .3, 1/ .9, 5/, .13, 43/, .13, 29/, .14, 50/, .14, 35/. From now it will be understood that

PC

2 C.Z/ and u.P / > 18.

(9.9)

108

Chapter 9 Bound for the coefficients of the linear form

We compute E.Q/ Š Z2 Z2 ZZ, so that the rank r D 2. The torsion subgroup consists of the points O and QiE D .ei , 0/ (i D 1, 2, 3). Thus, Etors .Q/ D hQ1E , Q2E i D h.113=12, 0/, .31=12, 0/i

&

r0 D 2.

As generators of the Z  Z part of the Mordell–Weil group we can take P1E D .79=12, 4/,

P2E D .67=12, 15=2/.

Let now g.u, v/ D 0, where u, v 2 Z and u > 18, and consider the point P C D .u, v/ 2 C.Z/. Then P E 2 E.Q/ and, specialising (4.2) to our case, we have P E D m1 P1E C m2 P2E C T E , implying

s L.P / D l.P / D .m0 C / !1 C m1 l.P1 / C m2 l.P2 /, 2 which is the specialisation of (4.6) to our case. By Theorem 9.1.2, L.P / D 0 implies m1 D m2 D 0, i.e. P E 2 Etors .Q/ D ¹Q1E , Q2E , Q1E C Q2E , Oº. Only for P D Q1 , Q2 we obtain a point P C with integer coordinates, namely Q1C D .9, 5/ and Q2C D .3, 1/, already listed in (9.9). Therefore, in what follows we assume that L.P / ¤ 0. The linear form L.P / falls under the scope of the first “bullet”, page 99, so that k D 2,

d D 1,

n0 D 2m0 C s, jsj  1,

.n1 , n2 / D .m1 , m2 /,

hence, N0 D 2M C 3 in (9.3) and in (9.6), .˛, ˇ/ D .2, 3/. Also, all points involved in the linear form have rational coordinates, therefore, in the notation of Preparatory to Theorem 9.1.2, page 103, D D 1. We also compute a fundamental pair of periods !1 0.8394944402534986

!2 1.0599189920498769  i ,

noting that D !2 =!1 1.2625682091829191  i satisfies = > 0, j j > 1 and j< j < 12 ; hence, in Lemma 9.1.1 we can take .$1 , $2 , Q / D .!1 , !2 , /. We calculate the canonical heights O 1 / 1.0547377027055837, h.P

O 2 / 0.3611721523399011 h.P

109

Section 9.3 Weierstrass equation example

and the height-pairing matrix   1.0547377027055837 0.2045874419576421 H 0.2045874419576421 0.3611721523399011 with least eigenvalue 0.6106414163983847. Now we apply Proposition 2.6.3. In the notation of that proposition, as a curve D we take the curve C of our example and, consequently, as and  therein, we respectively take 1 and 0. A straightforward calculation then gives  5.0391440734884212. Now we need to compute l.Pi / for i D 1, 2. Both PiE ’s belong to E1 .R/, therefore, from Conclusions and remarks (1), page 51, l.Pi / is the elliptic logarithm of the point PiE C Q2E which belongs to E0 .R/; more specifically, P1E C Q2E D .61=6, 51=4/,

P2E C Q2E D .173=12, 85=2/

and thus we compute `1 D l.P1 / 0.3588173091972342,

`2 D l.P2 / 0.2755755724720115.

Next we compute c12 6.4272819599100395  1028 , c14 D 1,

c13 9.1163192883638712  1073 ,

c15 25.402522816413709.

The remaining constants appearing in Theorem 9.1.3 are .c16 , c17 , c18 / D .23=2 ,

1 2

log 12, 1/.

Assuming M > c12 , we insert into (9.8) the values of ˛, ˇ,  , and c13 , : : : , c18 that we computed, and conclude that B.M / > 0, where B.M / D 9.11632  1073 .log.2M C 3/ C 1/¹log.log.2M C 3// C 25.402523º4 C 7.32132  0.61064M 2 . But, for all M > 1.1  1041 , we check that B.M / < 0, which implies that M  max¹c12 , 1.1  1041 º D 1.1  1041 .

(9.10)

We cannot obtain an upper bound for M essentially better than the above using Theorem 9.1.3; indeed, we check that B.1.07  1041 / > 0 which shows that with “a little smaller” bound for M we do not arrive at a contradiction. We will succeed to make an impressive reduction of the huge upper bound (9.10) in Section 10.2.1 of the next chapter.

110

Chapter 9 Bound for the coefficients of the linear form

9.4 Quartic equation example We use the notation, results etc. of Chapter 6. In (6.1) and (6.2) let us take a D 5=2, b D 1=2, c D 1, d D 0, e D 1, so that 5 1 C : 0 D Q.u/  v 2 D u4 C u3 C u2 C 1  v 2 . 2 2 By (6.3) we compute 685 31 , AD , BD 3 108 so that E : y 2 D f .x/ D x 3 C Ax C B. The approximate values of the roots of f .X / are e1 3.4860779869512022 > e2 0.6390559614483767 > e3 2.8470220255028254. We will work with the elliptic curve E, two models of which are C and E. The rank of E is 3 and the torsion subgroup Etors .Q/ is trivial; thus r D 3, r0 D 1. The following points form a basis for E.Q/: P1E D .2=3, 1=2/,

P2E D .5=3, 5=2/,

P3E D .8=3, 3=2/.

Let now .u, v/ be an integer solution of the equation Q.u/ D v 2 and P C D .u, v/ 2 C.Z/. Then P E 2 E.Q/ and, specialising (4.2) to our case, we have P E D m1 P1E C m2 P2E C m3 P3E implying l.P / D m0 !1 C m1 l.P1 / C m2 l.P2 / C m3 l.P3 /, which is the specialisation of (4.6) to our case. We also compute a fundamental pair of periods !1 1.3856089566101190,

!2 1.5968599642739782  i ,

with D !2 =!1 1.1524607694372033  i satisfying = > 0, j j > 1 and j< j < 12 , hence, in Lemma 9.1.1 we can take Q D .!1 , !2 , /. .$1 , $2 , / The approximate values of the canonical heights are O 1 / 0.55387253844458413, h.P O 2 / 0.6064693463824573, h.P O 3 / 0.6518823598427176 h.P

Section 9.4 Quartic equation example

111

and the height pairing matrix is 0 1 0.55387253844458413 0.0609518603595160 0.1563221441147483 H @ 0.0609518603595160 0.6064693463824573 0.1436252586206832 A 0.1563221441147483 0.1436252586206832 0.6518823598427176 with least eigenvalue 0.8549017536692952. Now we apply Proposition 2.6.3. In the notation of that proposition, as a curve D we take the minimal model of E which is D : y12 C y1 D x13 C x12  10x1  10. The change of variables relating D and E is x1 D x  13 , y1 D y  12 , therefore

D 1,  D  13 and from (2.41) we get  3.8640973065752320. We need to compute l.Pi / for i D 1, 2, 3. All three points PiE belong to E1 .R/, therefore, by Conclusions and remarks (1), page 51, it follows that l.Pi / is the elliptic logarithm of the point PiE C Q2E which belongs to E0 .R/ but does not have rational coordinates. Indeed, since f .X / is irreducible over Q and Q2E D .e2 , 0/, the coordinates of the points PiE C Q2E belong to the cubic number field Q.e2 /. Actually,   117173 1258 P1E C Q2E D 358 C 22e2  36e22 , C e2  656e22 18 3   1669 22 8 61477 472 532 2  e2  e22 , C e2  e2 P2E C Q2E D 225 15 25 2250 75 125   628 406 44 2 51653 10642 1256 2 E E  e2 C e2 , C e2  e . P3 C Q3 D  81 27 9 486 81 27 2 As noted in Section 9.2, the elliptic logarithm of points with irrational coordinates can be computed without problem. Thus, `1 D l.P1 / 0.0551123268560820, `2 D l.P2 / 0.3547306431725462, `3 D l.P3 / 0.5770693948785471. We will search for all points P on E such that P C 2 C.Z/. We will apply Theorem 6.8, which requires that ju.P /j be sufficiently large. Table 6.1 in Chapter 6 indicates how to compute how large ju.P /j should be. Specialising that table to our specific example we get Table 9.1. From Table 9.1 it follows that, with c7 D 3.498 in (6.15), Theorem 6.8 is certainly applicable to all points P 2 E such that .u.P /, v.P // are integers with v.P / > 0 and ju.P /j > 200. A quick computer search for points P with u.P / an integer of the interval Œ200, 200 reveals the following points: .u.P /, ˙v.P // D .130, 26701/, .4, 25/, .1, 2/, .0, 1/, .2, 7/. Therefore, from now on we will assume that ju.P /j > 200. We have L.P / D l.P / ˙ l.P0 /.

112

Chapter 9 Bound for the coefficients of the linear form

Table 9.1. Parameters and auxiliary functions for the solution of the quartic elliptic equation example (cf. Table 6.1). Q.u/ D 5 u4 C 1 u3 C u2 C 1

Q.u/ D 5 u4  1 u3 C u2 C 1

 D1

 D 1

2

2

u2 C 6 C 6  .Q.u//1=2 3u2 u0 D 1 u D 1 u D 200, c7 D 3.498 p P0E D . 1 C 10, 1 /

x.u/ D

3

2

x.u/ D

2

u  E

P0

2

u2 C 6 C 6  .Q.u//1=2 3u2 u0 D 17 u  D 17 D 17, c 7 D 3.496   p D 1 C 10,  1 3

2

cf. Lemma 6.1 and (6.11), (6.8) l.P0 /

l.P 0 / D l.P0 /

L.P / D l.P /  l.P0 /

L.P / D l.P / C l.P0 /

Using the routine IsLinearlyIndependent of MAGMA, we see that the points P0E , P1E , P2E , P3E are Z-linearly independent,11 so that we are in the situation described in the second “bullet”, page 99. Therefore, in (9.2), k D 4, d D 1, r0 D 1, .n0 , n1 , n2 , n3 / D .m0 , m1 , m2 , m3 /, n4 D ˙1, `4 D `0 . In (9.3) N0 D 32 M C

3 2

hence, in (9.6), .˛, ˇ/ D .3=2, 3=2/.

We compute `0 D l.P0 / 0.6737142222951107. O 0 /. Since P E is not a On applying Theorem 9.1.2, we will also need to compute h.P 0 rational point we confine ourselves to a reasonably good upper bound of its canonical height (see Section 9.2) which we obtain from Proposition 2.6.4. In the notation of 10x  10, which is, actually, that proposition we take D : y 2 C y D x 3 C x 2  p D thepminimal model of our elliptic curve. Then, P0 D . 10, 0/. By Proposition 2.4.2, h. 10/ D .log 10/=2 and then, by Proposition 2.6.4, O 0 /  4.12. h.P Finally, following the section Preparatory to Theorem 9.1.2 we see that D D 6. 11 For

a more transparent way to prove this fact see the end of this section.

113

Section 9.4 Quartic equation example

Indeed, in L.P / we have the elliptic logarithms of the points PiE C Q2E , the coordinates of which belong to Q.e2 /, and the elliptic logarithm of the point P0E , with p coordinates in Q. 10/. Therefore, the least number field containing the coordinates p of all these points is Q.e2 , 10/, the degree of which is 6. We also compute c12 9.5771564156066246  1021 ,

c13 5.6067135931158955  10165 ,

c14 2.7917594692280550,

c15 21.4114872494028374.

According to the announcement of Theorem 9.1.3, c16 2.5298221281347034,

c17 1.1754018326322600,

c18 D 1.

Assuming M > c12 , we insert into (9.8) the values of ˛, ˇ,  , and c13 , : : : , c18 that we computed, and conclude that B.M / > 0, where B.M / D 5.607  10165  ¹log.1.5M C 1.5/ C 2.79176º  ¹log.log.1.5M C 1.5/ C 21.4115º6 C 5.96764  0.8549  M 2 . But, for all M  2.3  1088 , we check that B.M / < 0, which implies that jM j  max¹c12 , 2.3  1088 º D 2.3  1088 .

(9.11)

We cannot obtain an upper bound for M essentially better than the above using Theorem 9.1.3; indeed, we check that B.2.2  1088 / > 0 which shows that “a little smaller” bound for M does not lead to a contradiction. A very big reduction of the huge upper bound (9.11) will be accomplished in Section 10.2.2 of the next chapter. Z-linear independence of P0E , P1E , P2E , P3E . If these points were Z-linearly deE 2 E.Q/. We show that pendent, then a positive integer d would exist, such that dP0p E this is impossible. Since P0 D .1=3 C , 1=2/, where  D 10, our claim is obvious for d D 1. Now we proceed by induction. Actually, we show something stronger: If RE D .m C n, p C q/ 2 E.Q.//, m, n, p, q 2 Q, then RE C P0E 62 E.Q/. We prove this as follows. From f .m C n, p C q/ D 0 we obtain 108m3 C 3240mn2  1116m  685  108p 2  1080q 2 D 0 1080n3 C 324m2 n  1116n  216pq D 0.

(9.12)

A symbolic computation gives  p10 .m, n, p, q/ C p11 .m, n, p, q/ , RE C P0E D d1 .m, n, p, q/  p20 .m, n, p, q/ C p21 .m, n, p, q/ , d2 .m, n, p, q/ where the pij ’s and di ’s are specific polynomials with integer coefficients. We have to show that p11 .m, n, p, q/ and p21 .m, n, p, q/ cannot both be zero. Suppose

114

Chapter 9 Bound for the coefficients of the linear form

the contrary and consider the two equations (9.12) along with the two equations p11 .m, n, p, q/ D 0 and p21 .m, n, p, q/ D 0. Elimination of q from these four equations leads to three polynomial equations in m, n, p with integer coefficients. A new elimination of p from the last three equations leads to two polynomial equations, say gi .m, n/ D 0, i D 1, 2 with integer coefficients. Specifically, g1 .m, n/ D n8 .9m2 C 30n2  31/6 h1 ,

g2 .m, n/ D n8 .9m2 C 30n2  31/3 h2 ,

where hi D hi .m, n/ 2 ZŒm, n, i D 1, 2. The resultant of h1 .m, n/ and h2 .m, n/ with respect to m belongs to ZŒn and its only rational roots are n D 1, 0, 1. For these values of n, the equation h1 .m, n/ D 0 has m D 1=3 as its only rational solution. Inserting into (9.12) .m, n/ D .1=3, 0/ gives an obviously impossible system in .p, q/; and inserting .m, n/ D .1=3, ˙1/ gives 27  108p 2  1080q 2 D 0 and pq D 0. If p D 0, then q 2 D 1=40, impossible for rational q; if q D 0, then p D ˙1=2 and RE D .m C n, p C q/ D .1=3 ˙ , ˙1=2/ is not a point of E. It remains to check what happens if 9m2 C 30n2  31 D 0. From the solution .m, n/ D .1=3, 1/ we find that all rational solutions to this equation are given by mD

10t 2  60t  3 , 3.10t 2 C 3/

nD

10t 2 C 2t  3 . 10t 2 C 3

(9.13)

From this, if we substitute for m and n in p11 .m, n, p, q/ D 0 we get an equation g.p, q, t / D 0; and if we substitute in (9.12) we get a system h.p, q, t / D 0, & pq D 0. Of course, g and h are polynomials with integer coefficients. If p D 0, then we have the equations g.0, q, t / D 0 and h.0, q, t / from which we eliminate q to obtain a polynomial in t with no rational solutions. If q D 0, we work similarly but this time we obtain a polynomial in t , the only rational root of which is t D 1=10. Inserting this value in (9.13) gives .m, n/ D .1=3, 1/. But this means that RE D ˙P0E and then RE C P0E is either the zero point O, which is excluded, or the point 2P0E , which has no rational coordinates. This completes the proof that no integer multiple of P0E has rational coordinates, without appealing to the routine IsLinearlyIndependent of MAGMA.

9.5 Simultaneous Pell equations example We use the notation, results etc. of Chapter 7. In (7.1) we take D1 D 11,

A1 D 14,

D2 D 7, A2 D 6. p p We check for which V D 1, : : : , 1000 both A1 C D1 V 2 and A2 C D2 V 2 are integers; only V D 1, 5 have this property, giving us the solutions .U , V , W / D .5, 1, 1/, .17, 5, 13/. As a particular solution .U0 , V0 , W0 / we choose .U0 , V0 , W0 / D .5, 1, 1/

115

Section 9.5 Simultaneous Pell equations example

and, from now on we assume that V > 1000. Consequently p p A1 C D1 V 2  3.317 V , A2 C D2 V 2  2.646 V

(9.14)

as required for the application of Theorem 7.1. By (7.12) and (7.13),  D C1,

.a, b, c, d , e/ D .36, 120, 376, 280, 14/,

Q.u/ D 36u4 C 120u3  376u2  280u C 196 and in (7.17) E : y 2 D x 3 C Ax C B D x 3 

326848 123412480 xC . 3 27

As noted a few lines after (7.28), f .X / has three rational roots, which are 1048 128 920 < e2 D < e1 D . 3 3 3 The functions u and v, defined in (7.18) and (7.19), are p 5V  14 C 11V 2 p , u.V / D V C  6 C 7V 2 p p 4.7  6 C 7V 2 C 82V  15 14 C 11V 2 / p . v.V / D .V C  6 C 7V 2 /2 e3 D 

Plotting v.V / and Q.u.V // for V > 1000 we see that v.V / and Q.u.V // are positive, as the second “bullet” in the announcement of Theorem 7.1 requires. Also, according to (7.21) and (7.22), p 28 Q.u/  280u C 392 376 C x.u/ D  3 u2 p 1680u3  10528u2  11760u C 10976 C .784  280u/ Q.u/ y.u/ D . u3 The functions V 7! x.V / and V 7! y.V / are defined in (7.23) by x.V / D .x ı u/.V /,

y.V / D .y ı u/.V /.

An explicit expression of x.V / and y.V / is an easy matter for a symbolic computation package, like MAPLE, for example, but this is a too long expression to print here. Besides, what we only need to check is (see third “bullet” in the announcement of Theorem 7.1) that, in the interval .1000, C1/ the function V 7! y.V / is positive and the function V 7! x.V / is strictly monotonous, with values exceeding e1 . Plotting

116

Chapter 9 Bound for the coefficients of the linear form

these two functions for V > 1000 we see that this is indeed true. More precisely, the function V 7! x.V / is strictly decreasing and its limiting value is p p p p 920 4164  2300 11 C 924 7  420 11 7 p > D e1 . 3  54 C 15 11 Finally, by (7.24) and (7.25) we compute P0E D .x0 , y0 /, where p p p p 1052 C 44 7 C 140 11 C 20 11 7 x0 D 3 p p p p p p y0 D 8 11 7 .5 7 C 11/.1 C 7/.5 C 11/. Now we calculate the constant c8 following the guidelines of the paragraph before the relation (7.32). From (7.31) we have x.V / D P1 .U , V , W /=P2 .U , V , W / D 2.840U C 4592V C 392W  479V 2 C 130U V C U 2  308V W C 140U W C 196W 2 /=.U  5V /2 . We obtain an upper bound for P1 .U , V , W / of the form constant times V 2 , where the constant can be obtained using (9.14) and V  0.001V 2 . We obtain a much better bound of this shape if we consider the function p p jP1 . 14 C 11V 2 , V ,  6 C 7V 2 /j V2 and observe that this is strictly decreasing in the interval .1000, C1/ taking a value < 3503.22 at V Dp1000. Working similarly for the denominator we see that p jP2 . 14 C 11V 2 , V ,  6 C 7V 2 /j=V 2 is a strictly increasing function with limiting value < 2.834 as V ! C1. Hence, we can take c8 D log.3503.22/. Now it is time to look for the arithmetic-geometric “details” of the model E. Our calculations have been made with a precision of 100 decimal digits; below we exhibit the numerical values in 16 digits. The free part of the group E.Q/ is generated by P1E D .1976=3, 14784/,

P2E D .1052=3, 3080/.

Also, Etors .Q/ Š Z2  Z2 is generated by Q1E D .e1 , 0/ and Q2E D .e2 , 0/, so that Etors .Q/ D h.920=3, 0/, .128, 0/i

&

r D 2, r0 D 2.

A fundamental pair of periods is !1 0.1520305707483552,

!2 D 0.1389333214997840  i

117

Section 9.5 Simultaneous Pell equations example

and, in the notation of Lemma 9.1.1, we take $1 D !2 ,

$2 D !1 .

For the point P E D .x.V /, y.V // coming from the unknown integer solution .U , V , W / of (7.1), with V > 1000 and U , W  0, we write P E D m1 P1E C m2 P2E C T E ,

T E 2 Etors .Q/,

so that (see immediately before relation (7.30)),   s L.P / D l.P / C l.P0 / D m0 C !1 C m1 l.P1 / C m2 l.P2 /  l.P0 /. 2 In view of (7.27), 2P0E D .1052=3, 3080/ D P2E , so that we are in the situation described in the third “bullet”, page 100. We compute that 2 l.P0 / D l.P2 /, therefore, L.P / D .m0 C s2 /!1 Cm1 `1 C.m2  12 /`2 , so that, in the notation of the third “bullet”, page 100, .d , d1 , d2 , ı0 , s0 =t0 / D .1, 0, 1, 0, 0=1/ and k D 2,

n0 D 2m0 C s, s 2 ¹1, 0, 1º,

.n1 , n2 / D .2m1 , 2m2  1/.

Therefore (see also (4.8)), jn0 j  2M0 C 1  2.M C 1/ C 1 D 2M C 3 and max1i2 jni j  2M C 1; hence, in (9.6) we can take .˛, ˇ/ D .2, 3/. Both points P1E and P2E belong to E0 .R/, therefore their l-values are equal to their respective elliptic logarithms. Thus, we calculate `1 D l.P1 / 0.0400052062019816,

`2 D l.P2 / 0.0606534846504600.

The canonical heights of P1 , P2 are O 1 / 0.7861708324460508, h.P

O 2 / 1.0456910246342577, h.P

and the height pairing matrix is   0.7861708324460508 0.4317829329642320 H 0.4317829329642320 1.0456910246342577 with minimum eigenvalue 0.9301430899748535.

118

Chapter 9 Bound for the coefficients of the linear form

We apply Proposition 2.6.3 in order to compute  . As a curve D we take the minimal model of E, which is D : y12 D x13  x12  6809x1 C 73689. The change of variables .x1 , y1 / D . 14 x C 13 , 18 y/ transforms D into E, hence, D 1=2,  D 1=3 and, according to Proposition 2.6.3,  5.4867659523958290. We also compute c12 1.4100217348795374  1035 , c14 D 1,

c13 1.6558700707656948  1074

c15 30.7739853633026035.

By the definition of c16 , c17 and c18 in Theorem 9.1.3, we obtain c16 0.0569802882298189,

c17 4.0807189122684443,

c18 D 1.

Assuming M > c12 , we insert into (9.8) the values of ˛, ˇ,  , and c13 , : : : , c18 that we computed, and conclude that B.M / > 0, where B.M / D 1.6559  1074 .log.2M C 3/ C 1/.log.log.2M C 3// C 30.774/4 C 6.702435  0.9301430  M 2 But, for all M  1.7  1041 , we check that B.M / < 0, which implies that jM j  max¹c12 , 1.7  1041 º D 1.7  1041 .

(9.15)

Since B.1.6  1041 / > 0, the bound (9.15) is essentially the best upper bound that can be obtained from Theorem 9.1.3; its reduction to an upper bound of manageable size will be accomplished in Section 10.2.3 of the next chapter.

9.6 General elliptic equation: A quintic example ExampleStep 7 . (continued from the end of Step 6, page 96) We have already computed c9 D 0.11 and  D 1=3 (see page 93), c10 D log.30000/ and c11 D 3 at Step 6, page 96, and P0E D .2169, 79947=2/ at Step 3, page 86. Therefore, by the definition of c16 , c17 and c18 we have c16 D 0.0825,

c17 D

1 2

log 3 C 2 log 10,

c18 D 92 .

The free part of E.Q/ is generated by the points P1E D .2439, 65853=2/,

P2E D .68292, 35626473=2/

and Etors .Q/ D ¹Oº. Moreover, P0 D P1 C P2 .

Section 9.6 General elliptic equation: A quintic example

119

A fundamental pair of periods is !1 0.1173285064197781, !2 0.0586642532098890 C 0.0183162358182653 i and we take $1 D !1 C 2!2 ,

$2 D !1 C !2 ,

Q D $2 =$1 0.5 C 1.6014276566418661 i . We have P0E 2 E.Q/; actually, P0E D P1E C P2E . At this point we note that f .X / (right-hand side of (8.5)) has only one real root, namely, e1 4898.1389881123821080 and, on calculating elliptic logarithms, we find out that l.P0 / D l.P1 /Cl.P2 /. Because of the relation P0E D P1E C P2E , our example falls in the case of the third “bullet”, page 100. Therefore (cf. Theorem 8.7.2), L.P / D l.P / C "P l.P0 / D m1 l.P1 / C m2 l.P2 / C m0 !1 C "P l.P1 C P2 / D m1 l.P1 / C m2 l.P2 / C m0 !1 C "P l.P1 / C "P l.P2 / D .m1 C "P /l.P1 / C .m2 C "P /l.P2 / C m0 !1 . By Theorem 3.5.2 (c), L.P / ¤ 0, except if .m1 , m2 / D ."P , "P /, i.e. only when P E D ˙P0E . This implies P D O, furnishing no solution to our Diophantine equation. Therefore, henceforth we assume L.P / ¤ 0 and we will apply Theorem 9.1.3 in combination with Theorem 8.7.2. In the notation of the third “bullet”, page 100, we have r0 D 1, s=t D s0 =t0 D 0=1, .d , d1 , d2 , d0 / D .1, 1, 1, 0/, k D 2, n0 D m0 and ni D mi ˙ 1 (i D 1, 2). Therefore, N D M0 D max0i2 jmi j D M C 1, by (4.8). Hence, in (9.6) we can take .˛, ˇ/ D .1, 1/. Next, we compute the canonical heights of P1 , P2 , O 1 / 0.2339011772221281, h.P

O 2 / 1.9162842498242523, h.P

and the height pairing matrix   0.2339011772221281 0.0519655844031980 H 0.0519655844031980 1.9162842498242523 with minimum eigenvalue 0.4645951770663837. Since f .X / has only one real root, we have E.R/ D E0 .R/ and, therefore, l.Pi / coincides with the elliptic logarithm of Pi for i D 1, 2. Thus, we compute `1 D l.P1 / 0.0296373863107625,

`2 D l.P2 / 0.0038280584521748.

120

Chapter 9 Bound for the coefficients of the linear form

Next, we apply Proposition 2.6.3 in order to compute  . As a curve D we take the minimal model of E, which is D : y12 C y1 D x13  220323x1 C 41292202. The 1 y  12 / transforms D into E. Then, applying change of variables .x1 , y1 / D . 19 x, 27 Proposition 2.6.3, we compute  5.4600863256138232. Finally, the values of the constants c12 , c13 , c14 and c15 are as follows: c12 1.31931  1030 ,

c13 1.042  1074 ,

c14 D 1,

c15 D 26.51416.

Now we have all data to apply Theorem 9.1.3. Inserting into (9.8) the values of the constants ˛, ˇ,  , and c13 , : : : , c18 which we have already computed, we see that B.M / > 0, where B.M / D 4.689  1074  .log.M C 1/ C 1/  .log.log.M C 1// C 26.5142/4  0.6127  0.46459  M 2 . But, for all M  3.1  1041 , we check that B.M / < 0, which implies that jM j  max¹c12 , 3.1  1041 º D 3.1  1041 .

(9.16)

Since B.3  1041 / > 0, the bound (9.16) is essentially the best upper bound that can be obtained from Theorem 9.1.3; its reduction to an upper bound of manageable size will be accomplished in Section 10.2.4 of the next chapter.

Chapter 10

Reducing the bound obtained in Chapter 9

In this chapter we will show how one can reduce the upper bound for M , which was obtained from the application of Theorem 9.1.3 and the Concluding remark following it. In our examples, more specifically, in Sections 9.3–9.6 we obtained an upper bound for M so large that there is no hope to check which points P “cover” solutions of the Diophantine problem under consideration; cf. relevant discussion on page 55, below (4.3). In the terminology therein, the Diophantine problems in our examples have been solved effectively but not explicitly. In the present chapter we aim to show how one can use the superficially “non-practical” huge upper bound for M in order to make an enormous “jump” down to a new really small upper bound. Sounds strange but, still, it is true! This new bound is so small that it allows one to check directly which points P “cover” actual solutions and leads provably to the complete set of them. We keep, of course, all notations etc. of Chapter 9; there we wrote L.P / for the linear form (9.2). Let us consider the following slightly different linear form, which is more appropriate for the purposes of the present chapter, dr0 L.P / D n0 C n1 1 C    C nk k , !1 r0 `i .i D 1, : : : , k/ i D !1

 D .P / D

(10.1)

The upper bound for jL.P /j given by (5.3), (6.15), (7.30) and (8.25) can be written 1 . C c  M 2 // (for c , c and c see Theouniformly as jL.P /j  c16 exp.c18 17 16 17 18 rem 9.1.3). Moreover, by (9.6)), N D max jni j  ˛M C ˇ, 0ik

so that M 2   3 N 2 , for some explicit positive constant 3 . It follows then that jj  1 exp. 2  4 N 2 /,

(10.2)

where

1 D

dr0 c16 , !1

1

2 D c18 . C c17 /,

and c16 , c17 and c18 are defined in Theorem 9.1.3.

1

4 D c18

3 .

(10.3)

122

Chapter 10 Reducing the bound obtained in Chapter 9

10.1 Reduction using the LLL-algorithm In this section we put our problem in a more general setting, totally independent from elliptic logarithms, as follows: Problem. Consider a linear form  given by (10.1), where 1 , : : : , k are known real numbers and n0 , n1 , : : : , nk are unknown integers. Suppose first, that an upper bound (10.2) holds for jj, where 1 , 2 and 4 are explicit positive constants, and second, def that a huge upper bound for N D max0ik jni j is known. Reduce this upper bound to a manageable size. We construct a lattice , which is a sublattice of ZkC1 . We will write the vectors of  as columns .k C 1/  1 and will denote them by bold letters. The lattice  that we will use in this section is generated by the column vectors of the matrix 1 0 1  0 0 B . . . C B . .. . .. C C, B . . M D B . C @ 0  1 0 A ŒC 1     ŒC k  C where C is a large positive integer which will be specified below. By Œx, where x 2 R, we mean the integer resulting from x if we cut its decimal digits; thus Œ3.85 D 3 and Œ3.85 D 3. Formally, Œx D dxe if x < 0, and Œx D bxc if x  0. We consider an LLL-reduced (ordered) basis .b0 , b1 , : : : , bk / of  in the sense of [27]. Roughly speaking, all vectors of an LLL-reduced basis have “approximately equal lengths” and are “approximately orthogonal” to each other. We explain a little more the important notion of an LLL-reduced basis, following [27, Section 1] (see also [72, Chapter 3] and [9, Section 2.6]) although these explanations are not necessary, in the strict sense of the word, for the applications of this book, in which we use only some properties of the LLL-reduced basis. We denote the usual euclidean length by j  j. Let .b0 , b1 , : : : , bk / be the (ordered) basis of RkC1 which we obtain if we apply to .b0 , b1 , : : : , bk / the Gram–Schmidt orthogonalisation process,1 so that, for 0  j < i  k, bi D bi 

i1 X

ij bj ,

j D1

ij D

hbi , bj i jbj j2

,

where hi denotes the usual inner product. The fact that .b0 , b1 , : : : , bk / is an LLLreduced basis means, by definition, that j ij j  1=2 1

.0  j < i  k/

b0 , b1 , : : : , bk is a basis of RkC1 but, in general, one cannot expect that it is a basis for .

123

Section 10.1 Reduction using the LLL-algorithm

and

jbi C i,i1 bi1 j2  34 jbi1 j2

.0 < i  k/.

Starting from any basis of a given lattice – in our case, from the basis consisting of the column vectors of M –, the LLL-algorithm of [27] computes an LLL-reduced basis very effectively; in our case, this means that the number of arithmetic operations that are needed are O..k C 1/4 log C /, in view of [27, Proposition 1.26]. LLL-reduced bases, in general, have a number of important properties, extremely useful for various applications, among them simultaneous approximation [27, Proposition 1.3] and factorisation of polynomials with rational coefficients [27, Sections 2 and 3]. The close connection of LLL-bases to the explicit solution of Diophantine equations was described for the first time2 in [72]. Applications to the explicit resolution of various classes of Diophantine equations are found, for example, in3 [64] (Thue equations), [65] (Thue–Mahler equations), [13] (norm form equations), [14] (“relative” Thue equations), [54, 15] (Weierstrass equations), [49] (Weierstrass equations over number fields), [62] (quartic elliptic equations), [59] (general cubic equations), [57] (elliptic binomial equations), [63] (simultaneous Pell equations), [56] (general elliptic equations), [48] (triangularly connected decomposable form equations), [47, 36] (S -integral solutions of Weierstrass equations); see also the book [50] for a concise overview of applications to Diophantine equations. An important property of the LLL-bases that we will use in this section is the following: Once we know b0 , we can have an explicit lower bound for the length of any non-zero vector of . More specifically, jxj  2k=2 jb0 j for every non-zero x 2 .

(10.4)

Suppose now that we want to solve the inequality (10.2), with  as in (10.1), where .n0 , n1 , : : : , nk / 2 ZkC1 and jni j < B1 .N / for i D 0, 1, : : : , k. Here we imagine B1 .N / as a “very large” integer. We consider the following x 2 : 1 1 0 0 1 0 n1 n1 n1 C B . C B . C B .. C B . C B . C B def C, C B B C B . x D M B . C D B . C D B C A @ nk A @ nk A @ nk n0 n1 ŒC 1  C    C nk ŒC k  C n0 C Q where Q D n1 ŒC 1  C    C nk ŒC k  C n0 C . For i D 1, : : : , k we have ŒC i  D C i C i , for some i absolutely less than 1, hence, Q D C   .n1 1 C    C nk k / and, consequently, Q  C jj C kN < C 1 e 2  4 N C kB1 .N /. jj 2

2 3

To the best of the author’s knowledge. Here we list, indicatively, only papers dealing with broad classes of Diophantine equations; the list is by no means exhaustive!

124

Chapter 10 Reducing the bound obtained in Chapter 9

Q and (10.2) we have By (10.4), the above estimate of jj k

2

2

jb0 j 

k X

2 jni j2 C Q 2  kB1 .N /2 C .C 1 e 2  4 N C kB1 .N //2 .

iD1

From this and a few elementary calculations we obtain q

4 N 2  2 C log. 1 C /  log¹ 2k jb0 j2  kB1 .N /  kB1 .N /º.

(10.5)

In order that the left-hand side makes sense, the quantity inside the brackets must be a positive real number, which is equivalent to 2k jb0 j2  kB1 .N /2 > k 2 B1 .N /2 . This, in turn, is equivalent to 2k jb0 j2 > .kB1 .N / C 12 /2  14 and, clearly, a sufficient condition for the last inequality is jb0 j > 2k=2 .k C 12 /B1 .N /.

(10.6)

The question is: How can we guarantee the validity of the last condition? Now is the moment for choosing C . Remember that the LLL-reduced basis of  is “almost orthogonal”. Heuristically, this implies that the volume of the parallelepiped formed by , : : : , bk – the fundamental volume of the lattice  – is “approxithe vectors b0 , b1Q mately” equal to kiD0 jbi j. But the volume of  is equal to the absolute value of the Q determinant of M , which is C , hence kiD0 jbi j is of the size of C . On the other hand, all vectors of an LLL-reduced basis have lengths of “same size”, hence we expect that jb0 j is of the size of C 1=.kC1/ . Therefore, if we had chosen the integer C somewhat larger than  1 kC1 B1 .N /kC1 , 013 (10.7) 2k.kC1/=2 k C 2 it would be reasonable to expect that condition (10.6) is fulfilled; if not, we would try a new choice for C , somewhat larger than the previous one; in practice this always works. Summing up, we have proved the following proposition: Proposition 10.1.1. Let  D n0 C n1 1 C    nk k , where 1 , : : : , k are specific real numbers and n0 , n1 , : : : , nk are unknown integers satisfying the two conditions N D max0ik jni j  B1 .N / and jj  1 exp. 2  4 N 2 /, where we understand that B1 .N / is a specific very large integer and 1 , 2 , 4 specific positive real numbers. Choose a positive integer C of size (10.7) and consider the lattice  defined at the beginning of this section. Compute an LLL-reduced basis .b0 , b1 , : : : , bk /. If b0 satisfies (10.6), then N satisfies (10.5). The very important result of this proposition is that the new upper bound for N is of the size of ² ³1=2    k 1 1=2 log 2 C log k C C log B1 .N / ,

4

2 C log 1 C .k C 1/ 2 2 p i.e. the size of the new upper bound for N is somewhat larger than log.B1 .N // !

125

Section 10.2 Examples

10.2

Examples

In this section we apply the reduction technique described in Section 10.1 to the linear form L.P / already obtained in each of Sections 9.3, 9.4, 9.5 and 9.6. In order to compute an LLL-reduced basis for the lattice  in each of our examples, we can use e.g. either of the following: 

The routine LLL of MAPLE with input data the column vectors of M .



The routine qflll of PARI with input data M and option 1, to declare that the routine must view its data as integers. Actually, qflll returns the transformation matrix from the reduced to the initial basis, so that the LLL-reduced basis is formed by the column vectors of the matrix product M  qflll.M /.



The routine LLL of matrix M .

MAGMA

with input data the matrix MT , the transpose of the

As b0 we can take either the first row of the matrix LLL.MT / (MAGMA), or the first column of the matrix M  qflll.M / (PARI), or the first vector of the vector array returned by LLL.Œc1 , c2 , c3 / (MAPLE), where c1 , c2 , c3 are the columns of M regarded as vectors.

10.2.1

Weierstrass equation

We will make use of some of the numerical results that we obtained in Section 9.3. We have k D 2, d D 1, r0 D 2, ˛ D 2, ˇ D 3 and we have computed all the parameters involved in the definition of 1 and 2 (see (10.3)). Concerning 3 , this must be chosen so that M 2  3 N 2 (cf. just above (10.2)). In our case, N  ˛M C ˇ D 2M C 3, 49 N 2 . For the right-most inequality it suffices to assume hence M 2  .N  3/2 =4  400 49 that N  10, which certainly holds if M  10. Thus, 3 D 400 and now, in view of (10.3), we can choose

1 D 6.73841,

2 D 6.2815,

4 D 0.0748.

In our case we have D

    2 2`1 2`2 L.P / D n0 C n1 C n2 D n0 C n1 1 C n2 2 , !1 !1 !1 D n0 C n1 .0.85484 : : :/ C n2 .0.65652 : : :/,

so that we have to compute !1 , `1 D l.P1 /, `2 D l.P2 / with a high precision. What is the precision required for our computations? According to (9.10), B1 .N / D 1.1  1041 and (10.7) suggests that the integer C be of the size of 1.66410125 . Let us be generous and choose C D 10130 . Since we must calculate ŒC i  (i D 1, 2), the choice of C forces us to make our computations with a precision of 130 decimal digits. For safety,

126

Chapter 10 Reducing the bound obtained in Chapter 9

we do them with a precision of 150 decimal digits. Then ŒC  is the integer resulting from C  if we truncate its decimal digits. In this way we calculate    0929152369 ŒC 1  D  8548414188 ƒ‚ …, „ 130 digits

and consider the sublattice  of

ŒC 2  D  6565274509    9728533696 ƒ‚ … „ 130 digits

Z3

generated by the columns of the matrix 0 1 1 0 0 1 0 A. M D @ 0 ŒC 1  ŒC 2  C

Then we compute an LLL-reduced basis of . It turns out that 0 1 5186369112644553909555354279275889624244640 b0 D @ 12154461378581626599733559868313990698122965 A . 8866073855682853407126296002425318856203365 We see that jb0 j 1.592  1043 which is of the size of C 1=3 2.1544347  1043 , in accordance with what we noted just above the relation (10.7). It is straightforward to check that b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N satisfies (10.5), which implies that N  52, an extremely good improvement! We set now B1 .N / D 52 and repeat the process, by choosing C D 108 , so that the new matrix M is 0 1 1 0 0 A 0 1 0 M D @ 85484141 65652745 100000000 and the LLL-reduction to the columns of the above matrix furnishes us with 0 1 117 new b0 D @ 166 A . 167 We check that (10.6) is satisfied so that, by Proposition 10.1.1 we have a new, even smaller, upper bound for N , obtained from (10.5). Indeed, the last relation implies that N  17, hence also M  17. Now we check which points P E D m1 P 1 C m2 P 2 C T ,

jm1 j, jm2 j  17,

T 2 Etors .Q/

have the property that P C has integer coordinates. In other words, given a point P E D .x, y/ as above, we check whether or not the point P C D .u, v/ with4 5 1 7 uDx , vDy x 12 2 24 has integer coordinates. A simple computer program can do the job very easily. Our computational results are collected in Table 10.1. 4

See the transformation at the beginning of Section 9.3.

127

Section 10.2 Examples Table 10.1. All points P E D n1 P1E C n2 P2E C T E with P C D .u, v/ 2 Z  Z. n1 , n2 , T E

P E D .x, y/

12 5 P C D .u, v/

1, 1, O 1, 0, O 1, 1, O 0, 2, O 0, 1, O 0, 1, O 0, 2, O 1, 1, O 1, 0, O 1, 1, O 1, 0, .41=6, 0/ 0, 1, .41=6, 0/ 0, 1, .41=6, 0/ 1, 0, .41=6, 0/ 1, 2, .31=12, 0/ 0, 3, .31=12, 0/ 0, 1, .31=12, 0/ 0, 0, .31=12, 0/ 0, 1, .31=12, 0/ 0, 3, .31=12, 0/ 1, 2, .31=12, 0/ 1, 1, .113=12, 0/ 0, 1, .113=12, 0/ 0, 0, .113=12, 0/ 0, 1, .113=12, 0/ 1, 1, .113=12, 0/

.293=12, 225=2/ .79=12, 4/ .1733=12, 3465=2/ .161=12, 36/ .67=12, 15=2/ .67=12, 15=2/ .161=12, 36/ .1733=12, 3465=2/ .79=12, 4/ .293=12, 225=2/ .3233=12, 4420/ .581=12, 663=2/ .581=12, 663=2/ .3233=12, 4420/ .2417=12, 2856/ .73613=12, 960925=2/ .173=12, 85=2/ .31=12, 0/ .173=12, 85=2/ .73613=12, 960925=2/ .2417=12, 2856/ .269=12, 195=2/ .43=12, 13=2/ .113=12, 0/ .43=12, 13=2/ .269=12, 195=2/

.24, 100/ .7, 1/ .144, 1805/ .13, 29/ .6, 5/ .6, 10/ .13, 43/ .144, 1660/ .7, 7/ .24, 125/ .269, 4285/ .48, 307/ .48, 356/ .269, 4555/ .201, 2957/ .6134, 477395/ .14, 50/ .3, 1/ .14, 35/ .6134, 483530/ .201, 2755/ .22, 109/ .4, 8/ .9, 5/ .4, 5/ .22, 86/

Proposition 10.2.1. All integer solutions .u, v/ of the equation v 2 C uv C v  u3  u2 C 71u C 196 D 0 are those listed in the right-most column of Table 10.1.

10.2.2

Quartic equation

In Section 9.4 we saw that k D 4, d D 1, r0 D 1, ˛ D 3=2, ˇ D 3=2 and we computed the parameters involved in the definition of 1 and 2 . Further, N  ˛M C ˇ D 3 .M C 1/, hence M 2  . 23 N  1/2  289 N 2 , if we assume M  10 (so that also 2 900 N  10). Thus, 3 D 289=900 and, in view of (10.3), we can take

1 D 1.8256,

2 D 5.0395,

4 D 0.274518.

128

Chapter 10 Reducing the bound obtained in Chapter 9

Choice of C : According to (9.11), B1 .N / D 2.3  1088 and (10.7) suggest that the integer C be of the size of 1.2162  10448 . We choose C D 10450 and ask our computer to work with a precision of 460 decimal digits. The linear form  to which we will apply the reduction process is         `1 `2 `3 `0 1 L.P / D n0 C n1 C n2 C n3 C n4 D !1 !1 !1 !1 !1 D n0 C n1 1 C n2 2 C n3 3 C n4 4 D n0 C n1 .0.03977 : : :/ C n2 .0.25601 : : :/ C n3 .0.41647 : : :/ C n4 .0.48622 : : :/. In the present case  is a sublattice of Z5 and 0 1 0 0 0 B 0 1 0 0 B 0 0 1 0 M D B B @ 0 0 0 1 ŒC 1  ŒC 2  ŒC 3  ŒC 4 

0 0 0 0 C

1 C C C, C A

where ŒC 1  D  3977480557    4553315116 ƒ‚ …, „

ŒC 2  D  2560106453    4697144898 ƒ‚ … „

ŒC 3  D  4164734877    6333939317 ƒ‚ …, „

ŒC 4  D  4862224793    9803530989 ƒ‚ … „

449 digits

450 digits

450 digits

450 digits

The first vector of the LLL-reduced basis is 0

1

b0 D

C A,

82327895474728509176238354944130475886404355810644437312762526348821779131423108393917712 B 15471125794154247726983556666031672982938641013672184463414407364168745466146688236207559 @ 111795263890064364945667733492720885363502478756971653430085800804576405203867180516532531 244350433420305070938288260659067994439387630719216219811147801739708020305288497923338381 491414271890840780959306036704856290884457656764343679435212935777287084290581917008213944

with approximate length 5.664  1089 , hence of the size of C 1=.kC1/ D 1090 , as expected. We check that b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N satisfies (10.5), which implies that N  55. We set now B1 .N / D 55 and repeat the process, by choosing C D 1016 . We obtain 0 1 151 B 77 C B C C new b0 D B B 378 C @ 62 A 984 which satisfies (10.6) and Proposition 10.1.1 asserts that we can apply relation (10.5). Doing so we obtain the new upper bound N  11, hence also M  11, and we check

129

Section 10.2 Examples

which points P E D m1 P 1 C m2 P 2 C m3 P 3 ,

jm1 j, jm2 j, jm3 j  11

have the property that P C has integer coordinates. In other words, we check for which points P E D .x, y/ as above, the point P C D .u, v/ with5   4.3x C 2/ 216x 3 C 216x 2  59  108y 2 C 108y .u, v/ D , (10.8) 3.2y  1/ 27.2y  1/2 has integer coordinates. Our computational results, obtained under the restriction y ¤ 1=2, are collected in Table 10.2. Table 10.2. All points P E D n1 P1E C n2 P2E C n3 P3 with P C D .u, v/ 2 Z  Z. n1 , n2 , n3 1, 1, O 1, 0, 1 1, 0, 0 1, 0, 1 1, 2, 0 0, 2, 0 0, 0, 1 0, 0, 1 0, 1, 0

12 5

P E D .x, y/ .19=3, 27=2/ .43=12, 13=8/ .2=3, 1=2/ .13=3, 11=2/ .1433=507, 2343=4394/ .262=75, 109=250/ .8=3, 3=2/ .8=3, 3=2/ .5=3, 5=2/

P C D .u, v/ .1, 2/ .4, 25/ .0, 1/ .2, 7/ .130, 26701/ .130, 26701/ .2, 7/ .4, 25/ .1, 2/

Does Table 10.2 miss any integer solution .u, v/? First, in view of 2y  1 appearing in the denominators of (10.8), we must check whether the rational point with y D 1=2, namely the point .2=3, 1=2/, corresponds to an integer solution .u, v/. Viewing (10.8) as an equality in the function field Q.E/, we see that the values of the functions u, v at the point .2=3, 1=2/ are 2=9 and 83=81, respectively.6 Actually, .u, v/ D .2=9, 83=81/ is a point on C but does not furnish an integer solution. Second, because the birational transformation C 3 .u, v/ 7! .x, y/ 2 E is given by7   6v C 6 C u2 8v C 8 C 4u2 C u3 .x, y/ D , , (10.9) 3u2 2u2 we must consider separately the value u D 0. Obviously we have the solutions .u, v/ D .0, 1/, .0, 1/, only the first of which is listed in the table. The reason why the solution 5 6 7

See (6.5) and (6.6). This is not so straightforward. A justification of this claim is given at the end of this section; see Singular values of the birational transformation. See (6.4).

130

Chapter 10 Reducing the bound obtained in Chapter 9

.u, v/ D .0, 1/ does not appear is because it corresponds to the zero point O 2 E, as a computation in the function field Q.C / shows.8 We have thus proved the following proposition: Proposition 10.2.2. All integer solutions of the equation 52 u4 C 12 u3 C u2 C 1 D v 2 are .u, v/ D .0, 1/ and those listed in the right-most column of Table 10.2. Singular values of the birational transformation. We must check the finitely many points .x, y/ 2 E, such that y is a zero of the denominator of the rational transformation E ! C separately, as to whether they correspond to a point on C which otherwise we might miss. In our example, we have to check only the point .x, y/ D .2=3, 1=2/. Below we explain why this point is mapped to the point .2=9, 81=83/ 2 C , hence to no integer solution of our Diophantine equation. The finitely many points .u, v/ 2 C , such that u is an integer and, at the same time, a zero of a denominator of the rational transformation C ! E can be trivially found. In this sense, it is not really necessary to find out to which points .x, y/ 2 E such points .u, v/ are mapped; nevertheless, below we discuss this issue as well. First, we work in the function field Q.E/ viewing u and v in (10.8) as functions. In view of y 2 D x 3 .31=3/x685=108 we have y 2 1=4 D .3xC2/.9x 2 6x89/=27, hence 27 2y C 1 3x C 2 D  2 2y  1 4 9x  6x  89 and, therefore, by (10.8), the value of the function u at the point .2=3, 1=2/ is u.2=3, 1=2/ D 2=9. Next, we consider the function v. By (10.8) we have   8 3x C 2 2 8.3x  1/.3x C 2/2  27.2y  1/2 D .3x  1/  1, vD 27.2y  1/2 27 2y  1 from which we see that v.2=3, 1=2/ D 83=81. This proves that, in the birational transformation from E to C , the point .x, y/ D .2=3, 1=2/ 2 E is mapped to the point .u, v/ D .2=9, 83=81/ 2 C . Next, we will show that, in the inverse transformation from C to E, the point .u, v/ D .0, 1/ 2 C is mapped to O 2 E. We work in the function field Q.C /. By v 2  1 D .5=2/u4  .1=2/u3 C u2 we have 5u2 C u C 2 vC1 D . u2 2.v  1/ Then, by this relation and (10.9), 1 15u2 C 3u C 5 C v vC1 C D , xD2 u2 3 3.v  1/ yD 8

4 vC1 2 1 2 5u2 C u C 2 2 1 20u2 C 3u C 4 C uv C 4v  2 C C D  C C D . u u u 2 u v1 u 2 2u.v  1/

See below Singular values of the birational transformation.

131

Section 10.2 Examples

Taking into account these expressions of x and y and writing projectively the functions u, v and x, y, using u D U=W , v D V =W and x D X=Z, y D Y =Z, we can express the rational map from the projective curve C to the projective curve E as follows:  .U : V : W / 7! 2.15U 2 C 3U W C 5W 2 C V W /U : 3.20U 2 C 3U W C 4W 2 C U V C 4V W /W :  6U W .V  W / , from which we see that the point .0 : 1 : 1/ 2 C is mapped to the point .0 : 1 : 0/ D O 2 E.

10.2.3

System of simultaneous Pell equations

In Section 9.5 we saw that k D 2, d D 1, r0 D 2 and ˛ D 2, ˇ D 3, so that, as in 49 N 2 , if we assume M  10; hence, 3 D 49=400. In Section Section 10.2.1, M 2  400 9.5 we also computed the parameters involved in the definition of 1 and 2 . Thus, in view of (10.3), we can take

1 D 0.7496,

2 D 9.5675,

4 D 0.1139.

Choice of C : According to (9.5), B1 .N / D 1.71041 and (10.7) suggest that the integer C be of the size of 6.14125  10125 . We choose C D 10126 and ask our computer to work with a precision of 140 decimal digits. The linear form  to which we will apply the reduction process is     `1 `2 1 L.P / D n0 C n1 C n2 D n0 C n1 1 C n2 2 D !1 !1 !1 D n0 C n1 .0.52627 : : :/ C n2 .0.79791 : : :/. The lattice  is a sublattice of Z3 and 0

1 1 0 0 1 0 A, M D @ 0 ŒC 1  ŒC 2  C

where ŒC 1  D  5262784452    0127775724 ƒ‚ …, „ 126 digits

ŒC 2  D  7979116877    4671848557 ƒ‚ … „ 126 digits

The first vector of the LLL-reduced basis is 0 1 1214390824826344155895547868148571092185 b0 D @ 637000902668796003574070357246352983001516 A . 641839507507851952420371897464942537470648

132

Chapter 10 Reducing the bound obtained in Chapter 9

The vector b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N satisfies (10.5), which implies that N  42. We set now B1 .N / D 42 and repeat the process, by choosing C D 108 . We obtain 0 1 96 37 A . new b0 D @ 192 The new b0 satisfies (10.6), hence we obtain the new upper bound N  14. Given a point .x, y/ 2 E we obtain a solution .U , V , W / to the system of simultaneous Pell equations as follows: 

From .x, y/ we obtain .u, v/ using (6.5) and (6.6), where  is as in (7.12), not as in Lemma 6.1.9



We use (7.15) to obtain the value of W and (7.16) to obtain V and U .

In this way we compute the following map .x, y/ 7! .U , V , W /: U D

9.5220x 2  4067520x C 642159360  132xy  46112y  15y 2 / , 908119808  166320y  28404x 2  27y 2 C 54x 3

VD

3.3924x 2  2197632x C 373490944 C 9y 2 C 180xy  7680y/ , 908119808  166320y  28404x 2  27y 2 C 54x 3

W D

3.8676x 2  2203968x  208626944 C 1260xy  386400y C 9y 2 / . 908119808  166320y  28404x 2  27y 2 C 54x 3

Then we check which points P , where P E D m1 P 1 C m2 P 2 C T ,

jm1 j, jm2 j  14,

T 2 Etors .Q/,

have the property that P E D .x, y/ is not a zero of the denominator of the above expressions of U , V , W and maps to an integer solution .U , V , W /. Our results are summarised in Table 10.3. The exceptional values of .x, y/. Now we must check the points .x, y/ 2 def E.Q/ which are zeros of the denominator q.x, y/ D 908119808  166320y  28404x 2  27y 2 C 54x 3 in the expressions of U , V , W above Table 10.3. Computing Resy .q.x, y/, f .x/  y 2 / we find out that it is the product of a quartic irreducible polynomial in x times .3x  1052/2 , hence the only points .x, y/ 2 E.Q/, for which q.x, y/ D 0, are .1052=3, ˙3080/.

9

But see the comment just before Lemma 6.1.

133

Section 10.2 Examples Table 10.3. All points P E D n1 P1E C n2 P2E C T E mapping to .U , V , W / 2 Z3 . n1 , n2 , T E

P E D .x, y/

12 3

1, 0, O 0, 1, O 1, 1, O 1, 0, .1048=3, 0/ 0, 1, .1048=3, 0/ 1, 1, .1048=3, 0/ 1, 0, .128=3, 0/ 0, 1, .128=3, 0/ 1, 1, .128=3, 0/ 1, 0, .920=3, 0/ 0, 1, .920=3, 0/ 1, 1, .920=3, 0/

P C D .u, v/

.1976=3, 14784/ .1052=3, 3080/ .1304=3, 6272/ .848=9, 101024=27/ .1352=75, 202048=125/ .64=3, 2624/ .376=3, 4032/ .880=3, 3360/ .664=3, 4224/ .2396=3, 20664/ .12728=3, 275520/ .4979=3, 66297/

.17, 5, 13/ .5, 1, 1/ .17, 5, 13/ .17, 5, 13/ .5, 1, 1/ .17, 5, 13/ .17, 5, 13/ .5, 1, 1/ .17, 5, 13/ .17, 5, 13/ .5, 1, 1/ .17, 5, 13/

We compute10 the Puiseux series of f .x/  y 2 around x D 1052=3. This is equiv/  y 2 around t D 0. We find alent to computing the Puiseux series of f .t C 1052 3 y1 .t / D 3080 

211 2603 2 tC t C O.t 3 / 5 22000

211 2603 2 t t C O.t 3 /. 5 22000 Inserting x D t C 1052=3, y D yi .t / (i D 1, 2) in the expressions of U , V and W and writing the resulting expressions as series in t , which we denote by Ui .t /, Vi .t /, Wi .t /, we obtain 18955 55389191 25665908007097 2  t t C O.t 3 / U1 .t / D  2603 948585260 76050356898824000 y2 .t / D 3080 C

V1 .t / D  W1 .t / D

4903 3893357 14906658779053  t t 2 C O.t 3 / 2603 189717052 167310785177412800

11297 18587273 3883710777647 C tC t 2 C O.t 3 / 2603 298126796 23901540739630400

and

10 All

U2 .t / D 5 

1 27 t t 2 C O.t 3 / 140 616000

V2 .t / D 1 

1 23 t t 2 C O.t 3 / 308 1355200

computations below were performed with MAPLE.

134

Chapter 10 Reducing the bound obtained in Chapter 9

W2 .t / D 1 

1 139 tC t 2 C O.t 3 /. 44 1355200

Thus, the solutions .U , V , W / corresponding to .x, y/ D .1052=3, ˙3080/ are .U1 .0/, V1 .0/, W1 .0// D 26031 .18955, 4903, 11297/ and .U2 .0/, V2 .0/, W2 .0// D .5, 1, 1/, respectively. The second solution is not essentially new, while the first, being a rational but not an integer solution to our system of Pell equations, is rejected. Thus, we have the following proposition: Proposition 10.2.3. All positive integer solutions of the simultaneous Pell equations U 2  11V 2 D 14, W 2  7V 2 D 6 are .U , V , W / D .5, 1, 1/, .17, 5, 13/.

10.2.4 General elliptic equation: A quintic example In Section 9.6 we saw that k D 2, d D 1, r0 D 1 and ˛ D 1, ˇ D 1, so that M 2  81 N 2 , if we assume M  10; hence 3 D 49=400. We have already computed in 100 Section 9.6 the parameters involved in the definition of 1 and 2 . Thus, in view of (10.3), we can take

1 D 0.7032,

2 D 2.3588,

4 D 0.0836.

Choice of C : According to (9.6), B1 .N / D 3.11041 and (10.7) suggest that the integer C be of the size of 3.724  10126 . We choose C D 10127 and work with a precision of 140 decimal digits. The linear form  to which we will apply the reduction process is     `1 `2 1 L.P / D n0 C n1 C n2 D n0 C n1 1 C n2 2 D !1 !1 !1 D n0 C n1 .0.25260 : : :/ C n2 .0.03262 : : :/. The lattice  is a sublattice of Z3 and 0

1 1 0 0 1 0 A, M D @ 0 ŒC 1  ŒC 2  C

where ŒC 1  D  2526017522 ƒ‚   97484775902 „ …, 128 digits

ƒ‚   70180087887 ŒC 2  D  3262684039 …. „ 127 digits

The first vector of the LLL-reduced basis is 0 1 2131974326945673073515451110976172071914863 763870092422929229257003034756651925844085 A . b0 D @ 1984748517852306190196449030197544381066969

135

Section 10.2 Examples

The vector b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N satisfies (10.5), which implies that N  48. We set now B1 .N / D 48 and repeat the process, by choosing C D 108 . We obtain 0 1 329 new b0 D @ 150 A . 175 The new b0 satisfies (10.6), hence we obtain the new upper bound N  13. We check, therefore, which points P , where P E D m1 P 1 C m2 P 2 ,

jm1 j, jm2 j  13,

have the property that P E D .x, y/ maps via the transformation (8.7) to a point .u, v/ 2 C with integer coordinates. The computer search does not include “singular” points .x, y/ that are zeros of the denominators in (8.7). For “non-singular” points our search is summarised in Table 10.4. Table 10.4. All points P E D n1 P1E C n2 P2E with P C D .u, v/ 2 Z  Z. 12 3 n1 , n2

P E D .x, y/

P C D .u, v/

0, 1 1, 2 1, 1

.68292, 35626473=2/ .182145033=58564, 969834918357=14172488/ .2710, 81029=2/

.0, 0/ .243, 3/ .0, 0/

The exceptional values of .x, y/. The irreducible factors of the denominators in the transformation (8.7) are x C 4365 and x  2169, hence we must check separately the points .x, y/ D .4365, ˙315171=2/, .2169, ˙79947=2/ 2 E.Q/. Instead of computing the Puiseux expansions of f .x/  y 2 around x D 4365, we compute the Puiseux series of f .t  4365/  y 2 around t D 0. We find y1 .t / D  y2 .t / D

315171 161784 1785281093 2  tC t C O.t 3 / 2 1297 19636425657

315171 161784 1785281093 2 C t t C O.t 3 /. 2 1297 19636425657

Then, by (8.7), U.t  4365, y1 .t // D 

13740972520149225 C O.t /, 23841086865992

V.t  4365, y1 .t // D 

6346275 C O.t /, 313874

136

Chapter 10 Reducing the bound obtained in Chapter 9

hence the point .x, y/ D .4365, 315171=2/ 2 E is mapped to the point .u, v/ D .13740972520149225=23841086865992, 6346275=313874/ 2 C , with non-integer coordinates. We compute also 492331523 C O.t / U.t  4365, y2 .t // D 1226330361 t 2  865647 t 1 C 1682209 16679 C O.t /, V.t  4365, y2 .t // D 35019 t 1 C 1297 which shows, if we write the rational map (8.7) projectively, that the projective point .4365 : 315171=2 : 1/ 2 E is mapped to the projective point .1 : 0 : 0/, the “point at infinity” of C , which does not furnish a solution to our Diophantine equation. Similarly, working with the points .2169, ˙79947=2/, we find that .2169, 79947=2/ is mapped to the non-integer point .19460428805563741=56309567282892, 1324909=238854/ 2 C , and .2169, 79947=2/ is mapped to the “point at infinity”. We have, thus, proved the following proposition: Proposition 10.2.4. The equation 3v 5 C 3uv 3  271uv  3u2 D 0 has .u, v/ D .0, 0/, .243, 3/ as its only integer solutions.

Chapter 11

S-integer solutions of Weierstrass equations

In this chapter we present a method for solving a Weierstrass equation in S -integers, a term that we immediately explain. Let an integer s  2 and S D ¹p1 , : : : , ps1 , 1º, where pi is a prime for i D 1, : : : , s  1 and 1 is merely a symbol. We often characterise 1 as the infinite prime, to distinguish it from the finite primes p1 , : : : , ps1 . The use of the infinity symbol 1 will be justified below. The systematic exposition of a method for the resolution of Weierstrass equations in S -integers is due to N. Smart [47] and Peth˝o–Zimmer–Gebel–Herrmann [36]; see also page x in the Preface. Let us agree that, for any non-zero rational number x, by denominator of x we mean the denominator b  1 of the irreducible fraction a=b D x (a, b integers) and by numerator of x we mean a. If the denominator of a rational prime x is divisible at most by primes in ¹p1 , : : : , ps1 º, then we say that x is an S -integer. We will include also 0 among the S -integers. The set of S -integers, which is obviously a subring of Q containing Z, will be denoted by ZS . The assumption card.S / D s  2 is made in order to ensure that S contains at least one finite prime; if this were not the case, then ZS coincides with Z and this chapter has nothing new to add to Chapter 5. Throughout this chapter, with the exception of Section 11.4, where a specific example is solved, S D ¹p1 , : : : , ps1 , 1º, s  2 where p1 , : : : , ps1 are finite primes. Our purpose is to present an explicit method for computing C.ZS /, where C : v 2 C a1 vu C a3 D u3 C a2 u2 C a4 u C a6 ,

a1 , a2 , a3 , a4 , a6 2 Z

(11.1)

is a model of an elliptic curve.

11.1

The formal group of C and p-adic elliptic logarithms

In this section we will study the curve C defined by (11.1) from the point of view of its formal group. Our exposition is based mainly on [45, Chapter IV]. We fix a (rational) prime p and, as usual, we denote by Qp the field of p-adic numbers (the completion of Q under the metric induced by the p-adic absolute value j  jp ) and by Zp the ring of p-adic integers, i.e. those x 2 Qp with jxjp  1. The

138

Chapter 11 S-integer solutions of Weierstrass equations

(only) maximal ideal of Zp is pZp which, for simplicity, we will denote by M. The natural homomorphism Zp ! Zp =M is denoted by t ! tQ. The residue field Zp =M is isomorphic to Fp D Z=pZ. The curve CQ =Fp defined by v 2 C aQ 1 uv C aQ 3 v D u3 C aQ 2 u2 C aQ 4 u C aQ 6 is the reduction of C mod p. The reduction mod p homomorphism C.Qp / 3 Q 7! QQ 2 CQ .Fp / is defined as follows. Any point Q 2 C.Qp / can be written in projective coordinates .U0 : V0 : W0 / with all three coordinates in Zp and not all zero. Then QQ D .UQ0 : VQ0 : WQ 0 / is a point belonging to CQ .Fp /. The curve CQ =Fp may have singular points but, anyway, the subset of the nonsingular points of CQ .Fp /, denoted by Cns .Fp /, is a group. We define C0 .Qp / D ¹Q 2 C.Qp / : QQ 2 Cns .Fp /º, which is a subgroup of C.Qp /, and the kernel of the reduction Q C1 .Qp / D ¹Q 2 C.Qp / : QQ D Oº; for the above we refer the reader to [45, beginning of Section VII.2 and Proposition 2.1]. Note that OQ 2 Cns .Fp /, hence C1 .Fp / C0 .Fp /. In order to define the formal group associated to C =Qp we need the model of (11.1) defined by def

w D z 3 C a1 zw C a2 z 2 w C a3 w 2 C a4 zw 2 C a6 w 3 D h.z, w/.

(11.2)

The birational transformation between (11.1) and (11.2) is  u 1 .u, v/ 7! .z, w/ D  ,  v v z 1 , . .z, w/ 7! .u, v/ D w w With a few obvious exceptions, any point Q of our elliptic curve can be viewed either as a pair .u.Q/, v.Q// satisfying equation (11.1) or as a pair .z.Q/, w.Q// satisfying equation (11.2), with the two pairs related by the above birational transformation. We characterise z.Q/ as the z-coordinate of Q. Lemma 11.1.1. If Q 2 C.Q/ \ C1 .Qp /, then z.Q/ 2 M. Proof. By Lemma 1.2.2, the coordinates of Q are .U=W 2 , V =W 3 /, where U , V , W are integers with gcd.U , W / D 1 D gcd.V , W /. Projectively, Q is the point ŒU W , V , W 3 , hence, the assumption QQ D OQ implies that W 0 .mod p/ and, consequently, U V 6 0 .mod p/. Then, z.Q/ D .U=W 2 / : .V =W 3 / D U W =V 2 M.

Section 11.1 The formal group of C and p-adic elliptic logarithms

139

If z 2 M and we define recursively h1 .z, w/ D h.z, w/ and

hmC1 .z, w/ D hm .z, h.z, w//,

then the limit w.z/ D lim hm .z, 0/ m!1

is expressed as a convergent in Qp power series in z and w.z/ D h.z, w.z//; see [45, Chapter IV, Proposition 1.1]. This makes it possible to express u and v as convergent power series of Qp as follows: z (11.3) D z 2  a1 z 1  a2  a3 z  .a4 C a1 a3 /z 2     2 Qp w.z/ 1 v.z/ D  D z 3 C a1 z 2 C a2 z 1 C a3 C .a4 C a1 a3 /z     2 Qp . w.z/

u.z/ D

The invariant differential1 then has the following expansion du.z/ !.z/ D 2v.z/ C a1 u.z/ C a3 D .1 C a1 z C .a12 C a2 /z 2 C .a13 C 2a1 a2 C 2a3 /z 3 C .a14 C 3a12 a2 C 6a1 a3 C a22 C 2a4 /z 4 C    /dz 2 Zp dz (see [45, Section IV.1]). For z1 , z2 2 M, a sum F.z1 , z2 / is defined by means of a p-adically convergent in M power series as follows:2 First, for z 2 M we define z z 2  a1 z 1     u.z/ D D 2 M. v.z/ C a1 u.z/ C a3 1 C a1 z C a3 w.z/ z 3 C 2a1 z 2 C    (11.4) Next we define w.z2 /  w.z1 / 2 M,  D .z1 , z2 / D w.z1 /  .z1 , z2 /z1 2 M,  D .z1 , z2 / D z2  z1 then i.z/ D

z3 D z3 .z1 , z2 / D z1  z2 

a1  C a3 2 C a2  C 2a4  C 3a6 2  2M 1 C a2  C a4 2 C a6 3

(the fact that ,  and z3 belong to M is not obvious at first glance; cf. [45, page 119]) and, finally, F.z1 , z2 / D i.z3 .z1 , z2 // D z1 C z2  a1 z1 z2  a2 .z12 z2 C z1 z22 / C .2a3 z13 z2 C .a1 a2  3a3 /z12 z22 C 2a3 z1 z23 / C    2 M. 1 2

[45, Chapter III]. [45, Section IV.1].

140

Chapter 11 S-integer solutions of Weierstrass equations

It is a fact3 that F.z1 , z2 / D F.z2 , z1 /, F.z1 , F.z2 , z3 // D F.F.z1 , z2 /, z3 / and F.z, i.z// D 0, hence the operation .z1 , z2 / 7! F.z1 , z2 / makes M an abelian group denoted CO .M/ or C; this is the formal group of C . For every integer  1, CO .M / is the subgroup of CO .M/ consisting of the elements of M . A remarkable property of the formal group is that, for any points Q1 , Q2 2 C.Q/ \ C1 .Qp /, we have4 z.Q1 C Q2 / D F.z.Q1 /, z.Q2 //. (11.5) Next, a logarithmic and an exponential function are defined. The (p-adic) logarithmic function on C D CO .M/ is defined5 by Z a1 a2 C a2 3 a13 C 2a1 a2 C 2a3 4 z C z C  logC z D !.z/ D z C z 2 C 1 2 3 4 1 X k k DzC (11.6) z 2 M, k kD2

with k 2 Z for every k. The above series is indeed convergent to an element of M; this follows from the fact that, for every k  2 it is true that p .z k =k/  k, which is an easy exercise to prove. ´ Let 1 if p > 2 D . (11.7) 2 if p D 2 

For z 2 M the logarithmic series is convergent and log z 2 M .

Since D 1 for p > 2, the above claim is already proven a few lines above; it suffices therefore to check for p D 2, when D 2. In this case, if z 2 M2 , then it is easily seen that 2 .z k =k/  k C 1 for every k  2, which is sufficient for the proof of the claim. The (p-adic) exponential function on C D CO .M/ is formally defined6 as the unique power series expC z satisfying logC .expC z/ D expC .logC z/ D z. P By [45, Chapter IV, Proposition 5.5], expC z D z C 1 kD2 every k. In analogy with the logarithmic function: 

kD2

is convergent and expC z 2 M . 4 5 6

zk ,

where k 2 Z for

For z 2 M the exponential series

1 X k k expC z D z C z , kŠ

3

(11.8) k kŠ

[45, page 120]. Remember that, by Lemma 11.1.1, z.Qi / 2 M for i D 1, 2. [45, Section IV.5 and Proposition 4.2]. [45, Section IV.5].

(11.9)

Section 11.1 The formal group of C and p-adic elliptic logarithms

141

To prove this, it suffices to show that the value of p at each summand of the infinite sum is at least . We have p .k t k =kŠ/  kp .t /  p .kŠ/  k  p .kŠ/. On the other hand,   k k k k k p .kŠ/ D C , C  < C 2 C  D 2 p p p p p1 k 1  k.  p1 /. The observation that the right-most hence, p .z k =kŠ/ > k  p1 side is  k=2 if p  3 and  k if p D 2 completes the proof of our claim. In view of the above discussion, we also conclude that:

If z 2 M then (11.8) is, indeed, meaningful as a relation in M . O a .M /, with as in (11.7), be the formal additive group, so that, by its Let now G O a .M / means usual addition in M . An important fact is definition, “addition” in G that the function O a .M / logC : CO .M / ! G 

is a group isomorphism, the inverse isomorphism being the function expC .7 This means in practice that, if z1 , z2 2 M , then logC F.z1 , z2 / D logC z1 C logC z2

expC .z1 C z2 / D F.expC z1 , expC z2 /. (11.10) We are almost ready to give the definition of the p-adic elliptic logarithm for points Q 2 C.Q/ \ C1 .Qp / which fulfil a certain condition. We give first a simple lemma useful here and in the sequel. and

Lemma 11.1.2. (i) Let Q D .u.Q/, v.Q// 2 C.Q/ and a prime p such that p .u.Q// < 0. Then, p .u.Q// is an even number and p .z.Q// D  12 p .u.Q//, hence jz.Q/jp D 1=2

ju.Q/jp . (ii) If Q D .u.Q/, v.Q// 2 C.Q/ and p .u.Q// < N for some positive N 2 Z, then, also, p .u.Q// < N . (iii) For i D 1, 2, let Qi D .u.Qi /, v.Qi // 2 C.Q/ be such that p .u.Qi // < N for some positive integer N . Then, p .u.Q1 C Q2 // < N . (iv) For i D 1, : : : , k, let Qi D .u.Qi /, v.Qi // 2 C.Q/ satisfy p .u.Qi // < N for some positive integer N . Then, p .u.m1 Q1 C    C mk Qk // < N for any integers m1 , : : : , mk . Proof. (i) In view of Lemma 1.2.2 we can write u.Q/ D U=W 2 and v.Q/ D V =W 3 , where U , V , W 2 Z and .U , W / D 1 D .V , W /. If p .u.Q// < 0, then, necessarily, p .W /  1 and, consequently, p .U / D 0 D p .V /. But then, p .u.Q// D 7

[45, Theorem 6.4, Chapter IV].

142

Chapter 11 S-integer solutions of Weierstrass equations

2p .W / which is an even number  4. Moreover, z.Q/ D u.Q/=v.Q/ D W U=V , hence p .z.Q// D p .W / D  12 p .u.Q//. (ii) We have z.Q/ D i.z.Q//, where i.z/ is the series (11.4). By (i), p .z.Q// D p .u.Q// > N=2 and now, if in (11.4) we put z D z.Q/, it becomes clear that p .i.z.Q/// > N=2, hence z.Q/ > N=2. On the other hand u.Q/ D u.Q/, therefore, applying (i), with Q in place of Q, we get p .u.Q// D 2p .z.Q// < N , as claimed. (iii) As in the proof of (ii), above, we see that p .z.Qi // > N=2 for i D 1, 2. By (11.5) we have z.Q1 C Q2 / D F.z.Q1 /, z.Q2 // and a look at the series F.z1 , z2 / on page 139 suffices to convince one that p .F.z.Q1 /, z.Q2 /// > N=2, hence p .z.Q1 C Q2 // > N=2 and then, by (11.3), p .u.Q1 C Q2 // D 2p .z.Q1 C Q2 // < N . (iv) Straightforward combination of (ii) and (iii). Suppose now that Q D .u.Q/, v.Q// is a point of C.Q/ with p .u.Q// < 0. Then, in view of Lemma 11.1.2(i), p .u.Q//  2. In the case that p D 2 we will make the stronger assumption that 2 .u.Q//  4. In other words, we assume that p .u.Q//  2 , where is defined in (11.7). Then, by Lemma 11.1.2(i), p .z.Q//  2 , hence, z.Q/ 2 M and, consequently, logC z.Q/ 2 M and expC .logC z.Q// D z.Q/. Definition 11.1.3. Let p be a prime and define by (11.7). If Q 2 C.Q/ and p .u.Q//  2 , then the p-adic elliptic logarithm of Q is, by definition, lp .Q/ D logC z.Q/ 2 M . Proposition 11.1.4. With p, Q as in Definition 11.1.3 we have jlp .Q/jp D jz.Q/jp . In particular, lp .Q/ ¤ 0. Proof. By hypothesis and Lemma 11.1.2(i), we can write u.Q/ D u1 =p 2t and v.Q/ D v1 =p 3t , where t  and p .u1 v1 / D 0. Then, z.Q/ D p t z1 , with p .z1 / D P 0. By definition, lp .Q/ D logC z.Q/ D p t z1 C k2 kk p kt z1k , where k 2 Z for all k. Since p .z.Q// D t , it suffices to show that k t  p .k/ > t for every k  2, which is an easy exercise (if p D 2 we must take into account that t  D 2). Finally, since jlp .Q/jp D jz.Q/jp D p t , it follows that lp .Q/ ¤ 0. Proposition 11.1.5. Let be defined by (11.7). If Q1 , : : : , Qk 2 C.Q/ such that p .u.Qi //  2 for i D 1, : : : , k, and n1 , : : : , nk are any integers, then z.n1 Q1 C    C nk Qk / D expC .n1 logC z.Q1 / C    nk logC z.Qk //,

(11.11)

or, equivalently, lp .n1 Q1 C    C nk Qk / D n1 lp .Q1 / C    nk lp .Qk /.

(11.12)

Section 11.1 The formal group of C and p-adic elliptic logarithms

143

Proof. The relation (11.11) is obtained by straightforward induction once we have proved the following: If Q1 , Q2 2 C.Q/ and p .u.Qi //  2 for i D 1, 2, then z.Q1 C Q2 / D expC .logC z.Q1 / C logC z.Q2 //. For the proof of this, let us put zi D logC z.Qi / (i D 1, 2). By our discussion just before Definition 11.1.3, z.Qi / 2 M , therefore, zi 2 M . Then, expC .logC z.Q1 / C logC z.Q2 // D expC .z1 C z2 / D F.expC z1 , expC z2 /, the right-most equality being implied by (11.10). We have F.expC z1 , expC z2 / D F.expC .logC z.Q1 //, expC .logC z.Q2 /// D F.z.Q1 /, z.Q2 // D z.Q1 C Q2 /, by (11.5), as claimed. Applying logC to the relation expC .logC z.Q1 / C logC z.Q2 // D z.Q1 C Q2 /, we obtain logC z.Q1 / C logC z.Q2 / D logC z.Q1 C Q2 /, which is equivalent to lp .Q1 / C lp .Q2 / D lp .Q1 C Q2 /. In our applications to Diophantine equations, points Q 2 C.Q/ will be involved which may not fulfil the condition p .u.Q//  2 necessary for both Definition 11.1.3 and relation (11.11). We overcome this complication using the fact that, for any point Q 2 C.Q/, there exists a positive integer m such that p .u.mQ//  2 ; see Proposition 11.1.7. First we prove a lemma. Lemma 11.1.6. Let Q D .u.Q/, v.Q// 2 C.Q/ and let p be a prime such that p .u.Q// < 0. (i) If m 2 Z and p .m/   12 p .u.Q//, then p .u.mQ//  2p .u.Q//. (ii) For any N 2 Z there exists an integer n such that p .u.nQ// < N . Proof. (i) By a1 , : : : , a6 in this proof we mean a1 , a2 , a3 , a4 , a6 . We will make use of various statements of Exercise 3.7 in [45], dealing with the so-called division polynomials k 2 ZŒu, a1 , : : : , a6  (k D 1, 2, : : :) and the related polynomials k and !k , again in 2 ZŒu, a1 , : : : , a6  . With the aid of these polynomials we can express the .u, v/-coordinates of the point kQ as   k .u.Q// !k .u.Q// , . kQ D 2 3 k .u.Q// k .u.Q// For the point Q in the announcement of the lemma, let us put for simplicity u.Q/ D u. Then, we have 2

m .u/ D um C .lower degree terms/ 2 ZŒa1 , : : : , a6 , u 2 m .u/

D m2 u m 2

2 1

C .lower degree terms/ 2 ZŒa1 , : : : , a6 , u.

It follows that p .um / is strictly less than the value of p at any other term of m .u/; therefore p .m .u// D m2 p .u/.

144

Chapter 11 S-integer solutions of Weierstrass equations

Now we estimate p . m .u/2 /. The value of p at any term different from the leading is at least .m2  2/p .u/, while p .m2 um

2 1

/ D .m2 1/p .u/C2p .m/  .m2 1/p .u/p .u/ D .m2 2/p .u/.

Therefore, p .

2 m .u/ /

 .m2  2/p .u/ and, consequently,

p .u.mQ// D p .m .u//  p .

2 m .u/ /

 m2 p .u/  .m2  2/p .u/ D 2p .u/.

(ii) For N  0 we choose n D m, where m 2 Z is as in (i). Next, let N < 0. In view of (i) we can choose m1 2 Z such that p .u.m1 Q//  2p .u.Q//. We put Q1 D m1 Q and apply once again (i) with Q1 in place of Q, finding m2 2 Z such that p .u.m2 Q1 //  2p .u.Q1 //. But then, p .m1 m2 Q/  22 p .u.Q//. Proceeding this way we find integers m1 , m2 , : : : , mk such that p .m1 m2    mk Q/  2k p .u.Q//. For sufficiently large k the right-hand side is < N and we take n D m1 m2    mk . Proposition 11.1.7. Let p be a prime and define by (11.7). For every point P 2 C.Q/ there exists a positive integer n such that p .u.nP //  2 . Consequently, by Definition 11.1.3, the p-adic elliptic logarithm l.nP / is meaningful. Proof. First we show that there exists a positive integer n0 , such that n0 PQ D OQ in the curve CQ =Fp . If the curve CQ =Fp is non-singular – hence it is an elliptic curve8 – we can take n0 D jCQ .Fp /j. If the curve CQ =Fp has singular points,9 then we turn to [45, Corollary 6.2, Chapter VII], according to which the subgroup C0 .Fp / of C.Fp / is of finite index, say k. Then f 2 Cns .Fp /. Since the reduction map is group homomorphism, kP 2 C.Fp /, hence, kP Q we obtain then k P 2 Cns .Fp /. If ` is the order of the finite group Cns .Fp /, then Q hence n0 PQ D OQ with n0 D `k. l.k PQ / D O, Q This means that p .u.Q// < 0. By Lemma Put Q D n0 P , so that QQ D O. 11.1.6, there exists a positive integer m such that p .u.mQ//  2p .u.Q//. By Lemma 11.1.2(i), p .u.Q//  2, hence p .u.mQ//  4  2 .

11.2 Points with coordinates in ZS In this and the following sections we will continue the study of the elliptic curve C defined by (11.1). For our numerical computations of p-adic elliptic logarithms, when p ¤ 1, we will make use of the routine pAdicEllipticLogarithm of MAGMA. This routine works with minimal models of elliptic curves, therefore, at this point we impose the further condition that C be a minimal model. This condition is not really restrictive, at least in principle. Indeed, let C0 : g0 .u0 , v0 / D 0 be a minimal model 8 9

In other words, if p is a prime of good reduction for C . In other words, if p is a prime of bad reduction for C .

Section 11.2 Points with coordinates in ZS

145

for C . The coordinates .u0 , v0 / are related to .u, v/ by .u0 , v0 / D . 2 u C , 3 v C

2 u C /, where , , ,  are appropriate rational numbers. If S0 is the set of finite primes dividing the product of the denominators of , , , , then, clearly, in order to compute all S -integer points .u, v/ on C , it suffices to compute all S [ S0 -integer points .u0 , v0 / on the minimal model C0 . Along with the model C we will need to consider a short Weierstrass model def

E : y 2 D f .x/ D x 3 C Ax C B,

A, B 2 Z

(11.13)

of the same elliptic curve, which we will denote by E. Obviously, everything in Section 11.1 applies if in place of the model (11.1) we have the model (11.13). A change of variable from C to E is of the form u D 2 x C ,

v D 3 y C 2 x C ,

(11.14)

with , , ,  appropriate rational numbers. The numbers , , and  can be chosen subject to the additional property that, in the inverse transformation .x, y/ D . 2 u   2 , 3 v  3 u C 3 .   // D . 0 2 u C 0 , 0 3 v C 0 u C 0 /,

(11.15)

the coefficients 0 , 0 , 0 , 0 are integers; in particular 1 2 Z. For example, with a2 C 4a2 a1 a3 C 4a1 a2  12a3 1 , D , D 1

D , D 1 6 12 2 24 we transform equation (11.1) into the equation (11.13), with A D 27a14  216a12 a2 C 1296a4  432a22 C 648a1 a3 , B D 3456a23 C 648a14 a2  7776a1 a2 a3 C 46656a6 C 11664a32  15552a4 a2  3888a4 a12  1944a13 a3 C 2592a12 a22 C 54a16 and .x, y/ D .36u C 3a12 C 12a2 , 216v C 108a1 u C 108a3 /. In specific numerical examples, we may find “better” values for , , and . Since in the inverse transformation from .u, v/ to .x, y/ the coefficients are integers, we see that, if .u, v/ 2 C.ZS /, then .x, y/ 2 E.ZS /. Notation. From now on, P will denote a typical point of E – the elliptic curve, two models of which are C and E – such that P C 2 C.ZS /, where S is the set of primes defined at the beginning of this chapter. In accordance with Section 11.1 we will write u.P / 1 , w.P C / D  , z.P C / D  v.P / v.P / x.P / 1 , w.P E / D  . z.P E / D  y.P / y.P /

146

Chapter 11 S-integer solutions of Weierstrass equations

As always, we denote the roots of f .X / by e1 , e2 , e3 , where e1 is real and e3 < e2 < e1 if all three roots are real. With , , ,  chosen as described immediately after (11.15) – in particular 1 2 Z – and as in (11.7), we define, for every prime q 2 S , bq D min¹1  2 C 2q . /, q ./, 2q . /, b1

´ 2 max¹je1 j, je3 jº D 2je1 j

2  . 3 q

 /º

if q ¤ 1

if e2 , e3 2 R . if e2 , e3 62 R

Note that, for q ¤ 1 we have bq < 0 because 1 2 Z and 2 ¹1, 2º. The two lemmas below are technical and will permit us to define a certain effectively computable10 finite subset S of C.ZS /; all points of C.ZS / outside S “behave properly” so that we can apply the theory of elliptic logarithms to the explicit computation of C.ZS /. The proofs of the lemmas are somewhat long, but elementary and not difficult at all. We note that the condition 1  2 C 2q . / for q ¤ 1 is not actually necessary for the proofs; it is only in Theorem 11.2.6 that we will need it. Lemma 11.2.1. (i) If q 2 ¹p1 , : : : , ps1 º and jx.P /jq > q 2q . /bq , then ju.P /jq D j j2q  jx.P /jq , jz.P E /jq D j jq  jz.P C /jq

jv.P /jq D j j3q  jy.P /jq , and

jz.P E /jq D jx.P /j1=2 . q

Moreover, both lq .P C / and lq .P E / exist and jlq .P E /jq D j jq jlq .P C /jq . (ii) If jx.P /j > b1 , then x.P / > 0, hence x.P / > b1 . Moreover, P E 2 E0 .R/ and Z 1 p dt  4 2  x.P /1=2 . p (11.16) 0< x.P / f .t / Proof. (i) As in Lemma 11.1.2(i), we write u.P / D U=W 2 and v.P / D V =W 3 , where U , V , W are integers and .U , W / D 1 D .V , W /. By hypothesis, q .x.P // > 2q . /  bq , hence q . 2 x.P // < bq  q ./ and then q .u.P // D q . 2 x.P / C / D q . 2 x.P // D 2q . / C q .x.P // < bq < 0. This implies that q .W / > 0 and q .U V / D 0, so that we can write u.P / D u1 =q 2ˇ , v.P / D v1 =q 3ˇ , where ˇ  1 and q .u1 v1 / D 0. Then, q .u.P // D 2ˇ and q .v.P // D 3ˇ. We now turn to (11.15), where 0 D 1 , 0 D  2 , 0 D  3 and 0 D 3 .   /. 10 Explicitly

computable, in practice, we hope!

Section 11.2 Points with coordinates in ZS

147

We have q .x.P // D q . 0 2 u.P / C 0 / and q . 0 2 u.P // < q .0 /. Indeed, the last inequality is equivalent to 2q . / C q .u.P // < q ./  2q . / which is true since, by hypothesis, q .u.P // < bq . Therefore, q .x.P // D q . 0 2 u.P // D 2q . / C q .u.P // D 2q . /  2ˇ and

ju.P /jq D q 2ˇ D q 2q . / q q .x.P // D j j2q  jx.P /jq .

Analogously, q .y.P // D q . 0 3 v.P / C 0 u.P / C 0 / and q . 0 3 v.P // is strictly less than both q . 0 u.P // and q .0 /, because of the relation q .u.P // < bq . For example, q . 0 3 v.P // < q . 0 u.P // is equivalent to 3q . / C q .v.P // < q . /  3q . / C q .u.P //, hence equivalent to ˇ > q . /, since q .u.P // D 2ˇ and q .v.P // D 3ˇ. We see that the last inequality is true on replacing ˇ by q .u.P //=2. We thus conclude that q .y.P // D q . 0 3 v.P // D 3q . / C q .v.P // D 3q . /  3ˇ and

jv.P /jq D q 3ˇ D q 3q . / q q .y.P // D j j3q  jy.P /jq ,

so that jz.P C /jq D

j j2q  jx.P /jq ju.P /jq E D D j j1 q  jz.P /jq . jv.P /jq j j3q  jy.P /jq

Finally, from ju.P /jq D j j2q  jx.P /jq we obtain 2q . / C q .x.P // D q .u.P // < bq ; hence, q .x.P // < bq  2q . /  0. Therefore x.P / D x0 =q 2t , y.P / D y0 =q 3t , where t  1 and q .x0 y0 / D 0. Consequently, jx.P /jq D q 2t and 1=2 jz.P E /jq D jx.P /=y.P /jq D jq t x0 =y0 jq D q t D jx.P /jq . C E Next we show that lq .P / and lq .P / are meaningful. By hypothesis, jx.P /jq > q 2q . /bq , which is equivalent to q .x.P // C 2q . / < bq . By the definition of bq we have bq  q ./, therefore, q .u.P // D q . 2 x.P / C / D q . 2 x.P //, because q . 2 x.P // D 2q . / C q .x.P // < bq  q ./. Thus, q .u.P // < bq  1  2 C 2q . /  1  2 (remember that 1 2 Z), hence q .u.P //  2 and we are allowed to define lp .P C / according to Definition 11.1.3. Also, we saw a few lines above that q .x.P // < bq  2q . / and, by the definition of bq , the right-hand side is  2 C 1; hence, q .x.P //  2 and, once again, Definition 11.1.3 allows us to define lq .P E / as well. Now, using Proposition 11.1.4, and the relation jz.P E /jq D j jq jz.P C /jq that we have already proved, we have jlq .P E /jq D jz.P E /jq D j jq jz.P C /jq D j jq jlq .P C /jq , as claimed. (ii) For simplicity in the notation, let us put x.P / D x. Note that, if e2 , e3 62 R, then x  e1 ; if e2 , e3 2 R, then either x  e1 , or e3  x  e2 . By hypothesis, jxj > b1 . First we show that x > 0. In the opposite case we would have x < b1 .

148

Chapter 11 S-integer solutions of Weierstrass equations

If e2 , e3 62 R, this means that x < 2je1 j  e1 , which is impossible. If e2 , e3 2 R, the inequality x < b1 means that x < 2 max¹je1 j, je3 jº and we distinguish two cases. If je1 j  je3 j, then x < 2je1 j  2je3 j, hence je3 j  e3  x < 2je1 j  2je3 j, which is impossible. If je1 j < je3 j, then x < 2je3 j, hence je3 j  e3  x < 2je3 j, again impossible. Now we know that x > b1 , in view of our discussion above. We will prove that f .x/ > 18 x 3 . Let first e2 , e3 2 R. From e3 < e2 < e1 we see that je2 j  max¹je1 j, je3 jº and now, from x > b1 , we obtain x > 2jei j  2ei for i D 1, 2, 3. It follows that x  ei > 12 x (i D 1, 2, 3) and f .x/ D .x  e1 /.x  e2 /.x  e3 / > 18 x 3 . Next, consider the case e2 , e3 62 R. As before, we have x  e1 > 12 x and we will show that .x  e2 /.x  e3 / > 14 x 2 . For this purpose we put e2 D t C wi with t , w 2 R and w ¤ 0, so that e3 D t  wi and e1 D 2t . Then .x  e2 /.x  e3 / D .x  t /2 C w 2 and it suffices to show that .x  t /2  14 x 2 , which is implied by x  t  12 x. The last inequality is true because x > 2je3 j > 2jt j  2t which we write as x  t > 12 x. From x.P / D x > b1 > 2je1 j  je1 j, we conclude in particular that P 2 E0 .R/. Note also that the real function t 7! f .t / is strictly increasing, so that f .t / > 18 t 3 for t  x.P /. Therefore, for any X > x.P /, we have Z

X

0< x.P /

p

dt f .t /

Z 

X x.P /

p 23=2 t 3=2 dt D 4 2  .x.P /1=2  X 1=2 /.

Letting X ! C1, we obtain (11.16). Lemma 11.2.2. For q 2 S we define ´ ¹P C 2 C.ZS / : maxp2S jx.P /jp D jx.P /jq  q 2q . /bq º if q ¤ 1 Pq D . ¹P C 2 C.ZS / : maxp2S jx.P /jp D jx.P /j1  b1 º if q D 1 Then, each set Pq is finite and, in principle, can be explicitly calculated. Proof. Since the coordinates of P C are S -integers and the coefficients 0 , 0 , 0 and 0 in (11.15) are integers, it follows that the coordinates of P E are also S -integers. Then, ˛s1 /, where, x0 2 Z and, for every i D 1, : : : , s 1, let us write x.P / D x0 =.p1˛1    ps1 it is true that ˛i  0 and the prime pi does divide x0 except, possibly, if ˛i D 0. (i) First, let q ¤ 1, so that q D pi0 2 ¹p1 , : : : , ps1 º. We put ˛ D ˛i0 . Applying the first statement of Lemma 11.2.3 below, with x.P / in place of x, we conclude that jx.P /jq  1 and then, necessarily, q .x.P // D ˛, hence jx.P /jq D q ˛ . In view of the hypothesis then, it follows that ˛  2q . /bq . If the right-hand side is < 0, then Pq D ;; therefore, let us suppose that 2q . /  bq  0. Then, for every i D 1, : : : , s  1, we have, jx.P /jpi  jx.P /jq D q ˛ , by the maximality of jx.P /jq . On the other hand, jx.P /jpi is D pi˛i , if ˛i  1, and  1, if

Section 11.2 Points with coordinates in ZS

149

˛i D 0. In any case, the relation jx.P /jpi  jx.P /jq implies that ± ° log q (i D 1, : : : , s  1). ˛i  max 0, .2q . /  bq / log p i

Thus, all exponents ˛i (i D 1, : : : , s  1) Q are bounded by explicit bounds. Finally, ˛i jx.P /j  jx.P /jq D q ˛ , hence jx0 j  q ˛ s1 iD1 pi , which provides us with an explicit upper bound for x0 . Existence of explicit upper bounds for the numerator and the denominator of jx.P /j means that we have an explicit upper bound for the Weil height of the point P E . Searching for all rational points of an elliptic curve with bounded Weil height is an effectively solvable problem, provided that the bound of the height is not very large; a bound for the Weil height of around 16 would require about an hour of computational time, using the routine ratpoints of M. Stoll. One can also use the routine Points of MAGMA. From the bounded set of points P E 2 E.ZS / that we compute this way, we easily recover the set Pq by means of (11.14). (ii) Next, let q D 1. Let P C 2 P1 . Now, for every prime pi (i D 1, : : : , s  1), we have jx.P /jpi  jx.P /j  b1 . Since jx.P /jpi is equal to pi˛i if ˛i > 0 and  1 if ˛i D 0, it follows that, anyway, ˛i  log b1 = log pi . Thus, the numerator and denominator of x.P / are bounded by explicit bounds hence, as already noted in the case q ¤ 1, finding all possibilities for P E is an effectively solvable problem. As noted in the case q ¤ 1, computing P1 is a trivial matter then. Notation. From now on and until the end of the chapter, we will consider the generic point P C D .u.P /, v.P // 2 C.ZS / X P, where [ PD Pp , (11.17) p2S

and the sets Pp are as in Lemma 11.2.2. For the corresponding point P E D .x.P /, y.P // 2 E.ZS / we follow the notation, assumptions etc. of Chapter 4 and write11 P E D m1 P1E C    mr PrE C T E ,

M D max jmi j. 1ir

(11.18)

Our main purpose is to obtain an upper bound for M . We will need to investigate Weil heights of points with S -integer coordinates. First, a few general remarks have their place. If x is a non-zero rational number and p is any (finite) prime, p divides the numerator of x if jxjp < 1, while p divides the denominator of x if jxjp > 1. Therefore, for any S -integer x, the formula (2.20) of the absolute logarithmic height of x becomes X log max¹1, jxjp º for x 2 ZS . h.x/ D p2S 11 Cf.

relation (4.2); all relations (4.5) through (4.8) are valid.

150

Chapter 11 S-integer solutions of Weierstrass equations

Also, Q the product formula (2.10) specialised to a non-zero rational x, takes the form p jxjp D 1, where we understand that p runs through all (finite) primes and 1. In the particular case that x is an S -integer we have Y Y Y jxjp  jxjp  jxjp , 1D p2S

p62S

p2S

where the right-most inequality holds because, if p 62 S , then jxjp  1, with strict inequality if p divides the numerator of x. The inequality above, in particular, proves the following simple but very useful lemma: Lemma 11.2.3. Let x be any S -integer and suppose that for some q 2 S we have jxjq D maxp2S jxjp . Then jxjq  1 and h.x/  s log jxjq . Remarks. (1) The isogeny of C and E and relation (11.18) imply that P C D m1 P1C C    mr PrC C T C .

(11.19)

(2) Combining Proposition 11.1.7 and Lemma 11.1.6 we see that for every q 2 S X ¹1º we can choose a positive integer tq satisfying q .u.tq Pi // < bq for every i D 1, : : : , r. Then, q .x.P // D q .u.P //  2q . / (cf. displayed relation at the beginning of the proof of Lemma 11.2.1(i)). Therefore, q .x.P // < bq  2q . /, which is equivalent to jx.P /jq > q 2q . /bq . Consequently, by Lemma 11.2.1(i), we conclude that, for every i D 1, : : : , r, the p-adic elliptic logarithms lq .PiC / and lq .PiE / are meaningful and satisfy jlq .P E /jq D j jq jlq .P C /jq . We will need to be more selective about our choice of tq because of our application of Theorem 11.2.5, which gives an explicit lower bound for a linear form in p-adic elliptic logarithms. In that theorem the number of elliptic logarithms involved in the linear form has an essential effect on the size of the bound; the smaller their number, the better the lower bound. Therefore, we will take tq with the extra property of being a multiple of the order of the Q-torsion subgroup, so that tq T is the zero point for every torsion point T . We have to impose one further condition on tq : The original paper [19] of N. Hirata-Kohno, of which Theorem 11.2.5 is an immediate corollary, requires, for convergence reasons, that jlq .tq PiE /jq < q  q , where q is defined in Theorem 11.2.5. Using Lemma 11.2.1(i), Proposition 11.1.4, and Lemma 11.1.2(i), we see that the last condition can be written as , q  q > j jq jlq .tq PiC /jq D j jq jz.tq PiC /jq D j jq ju.tq PiC /j1=2 q and this is equivalent to q .tq PiC / < 2q . /  2q . Summing up we have the following instructions for choosing tq : Choice of tq . If q ¤ 1 we choose the positive integer tq such that (i) q .tq PiC / < min¹bq , 2q . /  2q º for every i D 1, : : : , r, and (ii) tq T C D O for every torsion point T C 2 C.Q/.

Section 11.2 Points with coordinates in ZS

151

Theorem 11.2.4. Let P C 2 C.ZS / X P, where P is defined in (11.17). Let q 2 S be such that jx.P /jq D maxp2S jx.P /jp and choose tq as indicated above. Define ´ tq lq .P E / if q ¤ 1 def def and Lq .P C / D tq lq .P C / if q ¤ 1, Lq .P E / D l.P / if q D 1 where, in case q D 1, the map l is that of Theorem 3.5.2, and the integer m0 , satisfying jm0 j  12 rM , is chosen according to the discussion in Chapter 4 between relations (4.4) and (4.8). Then, the above definitions are meaningful, i.e. the q-adic elliptic logarithms lq .P C / and lq .P E / are meaningful, and Lq .P C / D m1 lq .tq P1C / C    C mr lq .tq PrC / ´ m1 lq .tq P1E / C    C mr lq .tq PrE / Lq .P E / D m1 l.P1 / C    C mr l.Pr / C l.T / C m0 !1

if q ¤ 1 if q ¤ 1 if q D 1.

Moreover, jLq .P E /jq  c19 exp.c20 M 2 /, where c19

´ jtq jq e =s D p =s 2 2e

if q ¤ 1 , if q D 1

(11.20)

c20 D =s

with  , defined in Propositions 2.6.2 and 2.6.3, respectively. In the case that q ¤ 1 we also have jLq .P C /jq D j 1 jq jLq .P E /jq .

(11.21)

Proof. Let P C 2 C.ZS / X P. We distinguish two cases. (i) Let q ¤ 1. Since P C 62 Pq , we have jx.P /jq > q 2q . /bq , which is equivalent to q .x.P // C 2q . / < bq . By the definition of bq we have bq  q ./, therefore, q .u.P // D q . 2 x.P / C / D q . 2 x.P //, because q . 2 x.P // D 2q . / C q .x.P // < bq  q ./. Thus, q .u.P // < bq  1  2 C 2q . /  1  2 (remember that 1 2 Z), hence q .u.P //  2 and we are allowed to define lq .P C / according to Definition 11.1.3. Also, we saw a few lines above that q .x.P // < bq  2q . / and, by the definition of bq , the right-hand side is  2 C 1; hence, q .x.P //  2 and, once again, Definition 11.1.3 allows us to define lq .P E /, as well. The relation (11.18) implies tq P E D m1 .tq P1E /C   mr .tq PrE /, hence, by Proposition 11.1.5 (in particular, relation (11.12)) Lq .P E / D tq lq .P E / D lq .tq P E / D m1 lq .tq P1E / C    mr lq .tq PrE /. Analogously, in view of (11.19), we obtain Lq .P C / D tq lq .P C / D lq .tq P C / D m1 lq .tq P1C / C    mr lq .tq PrC /.

152

Chapter 11 S-integer solutions of Weierstrass equations

Then, jLq .P E /jq D jtq jq jlq .P E /jq D jtq jq jz.P E /jq , by Proposition 11.1.4 and, analogously, jLq .P C /jq D jtq jq  jlq .P C /jq D jtq jq  jz.P C /jq . By Lemma 11.2.1, jz.P C /jq D j 1 jq jz.P E /jq , which proves the relation (11.21). Now we estimate jLq .P E /jq . We saw above that q .u.P // < bq , therefore, by 1=2 Lemma 11.2.1(i), jz.P E /jq D jx.P /jq . Thus, finally,  jtq jq  e h.x.P //=2s , jLq .P E /jq D jtq jq jx.P /j1=2 q

(11.22)

where the right-most inequality is implied by Lemma 11.2.3. Using Propositions 2.6.3 and 2.6.2 we obtain successively, 1 1 O //  1 .  M 2 /,  h.x.P //  .  h.P 2s s s therefore by (11.22), jLq .P E /jq  jtq jq e =s  e .=s/M , 2

as claimed. (ii) Let q D 1. The hypothesis P C 62 P1 implies jx.P /j > b1 so that, by relation (3.32) of Chapter 3 and Lemma 11.2.1(ii), we have Z p 1 C1 dt E  2 2 x.P /1=2 . jL1 .P /j1 D jl.P /j D p 2 x.P / f .t / By Lemma 11.2.3 we have h.x.P //  s log jx.P /j D s log x.P /. Then, log x.P /1=2  

1 h.x.P // 2s

1 O //  .  h.P s 1  .  M 2 / s

(by relation (5.1) of Proposition 2.6.3) (by Proposition 2.6.2).

Ifp we combine this with the above displayed upper bound of jl.P /j we obtain jl.P /j < 2 2 2e =s e .=s/M , as claimed. Finally, by definition, L1 .P E / D l.P / and, according to Chapter 4, relations (4.4) through (4.8), l.P / D m1 l.P1 / C    C mr l.Pr / C l.T / C m0 !1 , where m0 is an appropriate integer satisfying jm0 j  12 rM C 1. When we solve an elliptic Diophantine equation over Z, Theorem 9.1.3 was the basic tool for obtaining a first, very large, upper bound for M . Now that we are interested in solutions over ZS , the analogous tool is Theorem 11.2.6, below, which is heavily based on the following very important theorem.

Section 11.2 Points with coordinates in ZS

153

Theorem 11.2.5 (N. Hirata-Kohno). For 1 ¤ q 2 S consider the linear form Lq .P E / of Theorem 11.2.4 and define the following: h D log max¹jAj, jBjº, ´ 3 if q D 2 . q D 1=.q  1/ if q > 2 O i /º ai D max¹1, h, tq2 h.P H D .q q  max jlq .tq PiE /jq /1 , 1ir

.i D 1, : : : , r/, d D max¹1, 1= log H º

and g D max¹1, h, log a1 , : : : , log ar , log d º. If M 

eg ,

then jLq .P E /jq  exp.c23 log M /,

where c23 D 24r

2 C3r

.r C 1/2r

2 C9rC4

d 2rC2 g rC1 .log H /

(11.23) r Y

ai .

iD1

Proof. This theorem is an almost straightforward specialisation to our case of N. Hirata-Kohno’s result [19] which gives an effective lower bound for the p-adic absolute value of non-vanishing linear forms in p-adic elliptic logarithms. In the notation of that result, ui D lq .tq PiE / and expp .ui / D .x.Pi / : y.Pi / : 1/ (projectively). The nonvanishing of our linear form follows from its definition: Lq .P E / D tq lq .P E / ¤ 0 by Proposition 11.1.4. Theorem 11.2.6. With the assumptions and notations of Theorem 11.2.4, the following holds: 

Let q D 1. Referring to the model E in (11.13), follow the instructions in “Preparatory to Theorem 9.1.2 ”, page 103 to the linear form L1 .P E / in order to compute hE and H0 , H1 , : : : Hr , having in mind that, in the notation of the instructions, L.P / D L1 .P E / and in (9.2), k D r, d D 1, ni D mi for i D 0, 1, : : : , r, `i D l.Pi / for i D 1, : : : , r. Also, let and  be as in Propositions 2.6.2, and 2.6.3, respectively. Then, either def

˛M C ˇ  c12 D max¹exp.ehE /, jr0 j, exp.Hi /, i D 0, : : : , rº, where 12 ˛ D r0 max¹1, 12 rº, 12 We

ˇ D 32 r0 ,

remind that r0 is the lcm of the orders of non-zero points of Etors .Q/.

154

Chapter 11 S-integer solutions of Weierstrass equations

or M 2 < 32 s log 2 C  C c13 s.log.˛M C ˇ/ C 1/.log log.˛M C ˇ/ C 1 C hE /rC2 , (11.24) where r Y 2 2 Hi . c13 D 2.9  106rC12 42.rC1/ .r C 2/2r C13rC23.3 iD0 

Let q ¤ 1. If M  e g , then M 2  s log.jtq jq / C  C s c23 log M ,

(11.25)

where ,  are as in the case q D 1, tq is chosen as described just before the announcement of Theorem 11.2.4, s is the cardinality of S , and c23 is defined in Theorem 11.2.5. Proof. Let q D 1. Firstly, we claim that, if ˛M C ˇ > c12 , then   L1 .P E / > exp c13 .log.˛M C ˇ/ C 1/.log log.˛M C ˇ/ C 1 C hE /rC2 , (11.26) where c12 , c13 are those in the statement of the theorem. Our claim follows easily from Theorem 9.1.2 applied to the linear form L.P / D L1 .P E /. Indeed, as already noted in the announcement of the present theorem, in the notation of both Theorem 9.1.2 and relation (9.2), k D r, d D 1, ni D mi for i D 0, 1, : : : , r, and `i D l.Pi / for i D 1, : : : , r; also, for the N appearing in Theorem 9.1.2, we have N  ˛M C ˇ, according to the relation (9.6). For the explicit determination of ˛ and ˇ we refer to the first “bullet” on page 99 and we ascertain that, indeed, ˛ and ˇ are those in the statement of the present theorem. Now, (11.26) results from a straightforward application of Theorem 9.1.2, specialised to the present situation. It suffices to check (trivially) that c14 D 1, c15 D 1 C hE and c12 , c13 are those in the statement of the present theorem. On the other hand, by Theorem 11.2.4, L1 .P E /  c19 exp.c20 M 2 / with c19 D 23=2 e =s and c20 D =s. The combination of the last inequality with (11.26) immediately proves (11.24) in the case that ˛M C ˇ > c12 . Next, let q ¤ 1. Then, the relation (11.25) follows from a straightforward combination of the bounds (11.20) and (11.23). Remark. Obviously, both relations (11.24) and (11.25) imply an upper bound for M .

11.3 The p-adic reduction In this section, we assume that the assumptions of Theorem 11.2.6 hold for a certain finite prime q 2 S , which we will keep fixed until the end of the section. The reduction method that we expose is essentially due to B. M. M. de Weger [72].

155

Section 11.3 The p-adic reduction

For reasons explained at the very beginning of Section 11.2, we choose to work with the linear form Lq .P C /, rather than with Lq .P E /. On the other hand, the elliptic curve in N. Hirata-Kohno’s Theorem [19] is defined by a short Weierstrass model E with integer coefficients. Therefore we will have to pass from the model C to the model E and vice versa; in our transition from one model to the other, the basic tool is Lemma 11.2.1. Simplifying the notation, we rewrite the linear form Lq .P C / as follows: L D Lq .P C / D m1 1 C    C mk r , def

i D lq .tq PiC / .i D 1, : : : , r/.

By (11.20) and (11.21) it follows immediately that q .L/  c21 C c22 M 2 ,

log c19 , log q

c21 D q . / 

c22 D

c20 . log q

(11.27)

Forgetting now the specific meaning of the i ’s, except that they all belong to M , and the specific values of c21 and c22 , except that c22 > 0, we will solve the following computational problem: Problem. Let 0 , 1 , : : : , r 2 M and consider the linear form L D m1  1 C    C mr  r , where m1 , : : : , mr 2 Z. Let M D max1ir jmi j and assume the following: (i) M  Mq for some specific large bound Mq . (ii) q .L/  c21 C c22 M 2 , for some explicit parameters c21 and c22 > 0. (iii) (Without loss of generality) q .r / D min1ir q .i /. Under the assumptions (i)–(iii), find an upper bound of M considerably smaller than Mq . For any  2 Zq and any integer n > 1 we denote by  .n/ the n-th rational P approxii mation of . Thus,  .n/ 2 Z and q . .n/  / > n. In other words, if PD 1 iD0 ai q , n .n/ i where ai is an integer with 0  ai < p for every i  0, then  D iD0 ai q . First we consider the case r  2. We put i D  

r

LDƒD

r1 X iD1

i r

.i D 1, : : : , r  1/,

mi i C mr ,

.n/

ƒ

D

r1 X

.n/

mi i

C mr .

iD1 .n/

Note that, for i D 1, : : : , r  1 we have i 2 Zq and, consequently, i

2 Z.

156

Chapter 11 S-integer solutions of Weierstrass equations

For every integer n  1 we consider the lattice  .n/ generated by the column vectors of the matrix 0 1 1  0 0 B . .. .. C B . .. C .n/ B . . . . C Z DB C. @ 0  1 0 A .n/

.n/

   r1 q n

1 Proposition 11.3.1.

0 q .ƒ/  n

,

1 m1 B . C B . C 2  .n/ . @ . A mr .n/

Proof. Suppose first that q .ƒ/  n. From q .i  i / > n for every i D 1, : : : , r, we see that q .ƒ  ƒ.n/ /  n, hence q .ƒ.n/ /  n. Since ƒ.n/ 2 Z, it follows that P .n/ C mr D xr q n for some xr 2 Z. Consequently, ƒ.n/ D  r1 iD1 mi i 0 1 0 1 m1 m1 B . C B . C B . C B . C C B C Z .n/ B B . C D B . C, @ mr1 A @ mr1 A xr mr which shows that the right-hand side belongs to  .n/ . Conversely, if 1 0 m1 B . C B . C 2  .n/ , @ . A mr then, there exist x1 , : : : , xr1 , xr 2 Z such that 0 1 0 x1 m1 B . C B . B . C B . C B Z .n/ B B . CDB . @ xr1 A @ mr1 xr mr

1 C C C. C A

P .n/ C xr q n . ConseIt follows that mi D xi for i D 1, : : : , r  1 and mr D r1 iD1 xi i quently, r1 r1 X X .n/ .n/ ƒ.n/ D  xi i C . xi i C xr q n / D xr q n , iD1

iD1

157

Section 11.3 The p-adic reduction

which shows that q .ƒ.n/ /  n. Since q .ƒ  ƒ.n/ /  n, this proves that q .ƒ/  n. Theorem 11.3.2. With the assumptions of the Problem at the beginning of this section, the following is true.pLet b1 be the first vector of an LLL-reduced basis of the lattice Z .n/ . If jb1 j > 2r=2 r Mq then M2 

n  1 C q .r /  c21 . c22

Proof. Suppose that M 2 > .n  1 C q .r /  c21 /=c22 . Then, q .ƒ/ D q .L/  q .r /  c21 C c22 M 2  q .r / > n  1, hence q .ƒ/  n. Then, by Proposition 11.3.1, 1 0 m1 B . C B . C 2  .n/ . @ . A mr But the norm of every point of  .n/ is at least 2r=2 jb1 j, by [27, Proposition (1.11)]; therefore q p p p r M  m21 C    C m2r  2r=2 jb1 j > 2r=2 .2r=2 r Mq / D rMq , which contradicts the assumption M  Mq . Now we consider the case r D 1. In this case we have the following situation: L D 2 m1 1 , M D jm1 j and jLjq < q c21 c22 jm1 j . Since m1 2 Z, we have jm1 j1  jm1 jq , therefore, q .1 /c21 c22 jm1 j2 , jm1 j1  jm1 jq D jLjq j1 j1 q log q  .c21  q .1 / C c22 jm1 j2 /. This inequality, obviously, implies an upper bound for jm1 j. Since we expect that the positive parameter c22 is not extremely small, the upper bound that we obtain is so small that no further reduction of it is really necessary. .n/ Remark. How large should we p choose n when we construct the matrix Z , in order r=2 r Mq of Theorem 11.3.2 be fulfilled? Heuristically, we that the condition jb1 j > 2 argue as follows. Since the r vectors of the LLL-reduced basis have “almost equal” lengths and are “almost orthogonal” to each other, the volume of the parallelepiped formed by them is approximately jb1 jr . On the other hand, this volume is equal to

158

Chapter 11 S-integer solutions of Weierstrass equations

the volume formed by the initial basis of the lattice which, in turn is equal to the determinant of Z .n/ . But this is q n , thereforepwe expect that jb1 j is of the size of q n=r . Since we want that jb1 j satisfies jb1 j > 2r=2 r Mq , we must choose n so that & ' p r log.2r=2 rMq n . (11.28) log q

11.4 Example We will compute all S -integral points on C : v 2 C v D u3  7u C 6,

(11.29)

for S D ¹2, 3, 5, 7, 1º. Note that C is a minimal model. The rank over Q is 3 and the torsion subgroup is trivial. A Mordell–Weil basis is P1C D .1, 0/,

P2C D .2, 0/,

P3C D .0, 3/

and an isomorphic to C short Weierstrass model is E : y 2 D x 3  112x C 400, obtained by means of the transformation .u, v/ D . 14 x, 18 y  12 /. Thus, in the notation of (11.14), 1 1

D ,  D 0, D 0,  D  2 2 and P1E D .4, 4/, P2E D .8, 4/, P3E D .0, 20/. In Proposition 2.6.3, in place of the curve D we take our present curve C and we compute  2.87085138. Also, applying Proposition 2.6.2, we obtain 0.48599751, therefore c20 D 3=5. We need  and for calculating the parameters c19 (depending on the prime q 2 S ) of Theorem 11.2.6. First we compute all points P C 2 P; cf. Lemma 11.2.2. We compute b1 24.09944 < 24.1 and .b2 , b3 , b5 , b7 / D .5, 1, 1, 1/. Following the steps of the constructive proof of Lemma 11.2.2, it is easy to see that, if P C 2 P, then the height of x.P / (maximum of the numerator and denominator of jx.P /j) is less than 121462. Using the routine Points of MAGMA, we explicitly compute 274 points only 119 of which have S -integral coordinates; thus P is a subset of an explicit set P consisting of 119 S -integral points.

159

Section 11.4 Example

Now we assume that P C 62 P so that we can apply Theorem 11.2.6. We denote by q that prime in S for which jx.P /jq D maxp2S jx.P /jp and we distinguish cases according to q D 2, 3, 5, 7, 1. In the case that q ¤ 1, the routine pAdicEllipticLogarithm of MAGMA is used for computing the lq -values. 

q D 2. The instructions for the choice of tq on page 150 lead us to t2 D 80, so that L2 .P C / D m1  l2 .80P1C / C m2  l2 .80P2C / C m3  l2 .80P3C /. Theorem 11.2.6, in particular relation (11.25), provides us with the upper bound M  M2 D 1029 . Now we reduce this large upper bound M2 . We need first to calculate the parameters c21 and c22 according to (11.27); we actually need an approximation (not a high one) from below: c21 3.17164739, c22 0.865617. According to (11.28) we must work with a 2-adic precision at least O.2316 /. We prefer to work with a somewhat larger precision, choosing n D 320 in order to construct the matrix Z .n/ . A first low precision computation shows that 2 .l2 .80PiC // D 6 for i D 1, 2 and 2 .l2 .80P3C // D 5. Therefore we put i D l2 .80PiC / for i D 1, 2, 3, so that i D i =3 for i D 1, 2. Using MAGMA we compute .320/

1

D„ 224782ƒ‚    369609 … 2, 96 digits

.320/

2

D 163742    694465 „ ƒ‚ … 2. 96 digits

In the notation of Section 11.3 we have 0 1 1 0 0 1 0 A Z .320/ D @ 0 .320/ .320/ 320 2 2 1 and we compute an LLL-reduced basis for the lattice  .320/ generated by the vectorcolumns of Z .320/ .13 The first vector of the reduced basis is 0 1 28345891354399044132619085347326 b1 D @ 87234034300570141766432578346775 A . 7183933776826462841800336346962 p Its length is approximately 9.2004  1031 , “slightly larger” than 23=2 3 M2 4.89898  1031 , hence b1 satisfies the condition of Theorem 11.3.2. Consequently, according to the theorem, M 2  .321 C 2  c21 /=c22 < 370, therefore M  19. Repeating the reduction process with M2 and 2-adic precision O.222 /, we reduce the upper bound for M further, finding M2 D 4. 

q D 3. The P instructions for the choice of tq on page 150 lead us to t3 D 7, so that L3 .P C / D 3iD1 mi  l3 .7PiC /.

13 See

the “bullets” on page 125.

160

Chapter 11 S-integer solutions of Weierstrass equations

Theorem 11.2.6, in particular relation (11.25), provides us with an upper bound M  M3 D 1028 . For the reduction of M3 we calculate c21 0.52263230 and c22 0.546143.



We compute 3 .l3 .7PiC // D 1 for i D 1, 3 and 3 .l3 .7P2C // D 2. Therefore we put i D l3 .7Pi / for i D 1, 2, 3, so that i D i =3 for i D 1, 2. Then we work as in the case q D 2. With 3-adic precision O.3189 /, we reduce the upper bound for M , finding M  M3 D 18. One further reduction step with precision O.315 / leads to M  M3 D 5. P q D 5. Now we choose t5 D 10, so that L5 .P C / D 3iD1 mi  l5 .10PiC /. The relation (11.25) implies upper bound M  M5 D 1028 . For the reduction of M5 we calculate c21 0.6432479 and c22 0.3728.



We compute 5 .l5 .10PiC // D 2 for i D 1, 3 and 5 .l5 .10P2C // D 3. Therefore we put i D l5 .10PiC / for i D 1, 2, 3, so that i D i =3 for i D 1, 2. Then we work as in the case q D 2. With 5-adic precision O.5128 /, we reduce the upper bound for M , finding M  M5 D 18. One further reduction step with precision O.511 / results to M  M5 D 5. P q D 7. Now we choose t7 D 6, so that L7 .P C / D 3iD1 mi  l7 .6PiC /. The relation (11.25) implies the upper bound M  M7 D 1027 . For the reduction of M7 we calculate c21 0.29506515 and c22 0.308339. We compute 7 .l7 .6PiC // D 2 for i D 1, 2 and 7 .l7 .6P3C // D 3. Therefore we put 1 D l7 .6P3C /, 2 D l7 .6P2C / and 3 D l7 .6P1C /, so that i D i =3 for i D 1, 2. Working as before with 7-adic precision O.7104 /, we find in the first reduction step M  M7 D 18 and one further reduction step with precision O.79 / results to M  M7 D 5.



q D 1. According to Theorem 11.2.6, we work with the linear form L1 .P E / D m1 l.P1 / C m2 l.P2 / C m3 l.P3 / C m0 !1 , where, as always, !1 is the least positive real period for the corresponding Weierstrass } function. We calculate !1 1.03792199 and, in the notation of Lemma 9.1.1, we take $1 0.74027413, $2 D !1 , so that Q 1.40207788i . Further, we compute (note that, in our example, r0 D 1, hence ˛ D 3=2 D ˇ) c12 3.99858611020 , c13 6.651560410110 , c19 5.02231532 and c20 0.09719950. According to Theorem 11.2.6, if M > c12 then it satisfies the inequality (11.24) and an easy computation shows that this is not possible if M  2.6  1060 , hence we obtain the upper bound M  M1 D 2.6  1060 . We reduce M1 using an LLLreduction process completely analogous to that in the examples of Section 10.2, therefore we do not think it worthwhile to give any numerical details. We only say

161

Section 11.4 Example

that, in the first reduction step, the upper bound falls to M1 D 51 and the next step reduces it further to M1 D 13. As expected, all 119 points in P (see the beginning of this section) are rediscovered by our search, while 9 points from those found do not belong to P. Summing up, we conclude that M  max¹M2 , M3 , M5 , M7 , M1 º D 13 and we check all points P C D m1 P1C C m2 P2C C m3 P3C with M D max1i3 jmi jº for S -integrality. The results of our search is summarised in Table 11.1. Table 11.1. All S -integral points P C D m1 P1C C m2 P2C C m3 P3C on the curve (11.29), S D ¹2, 3, 5, 7, 1º.

m1 , m2 , m3 4, 2, 0 4, 0, 1 3, 2, 0 3, 2, 2 3, 1, 1 3, 1, 3 3, 0, 1 3, 1, 0 3, 1, 1 2, 3, 2 2, 3, 2 2, 2, 0

PC  1541761

,  1882859327 153664 60236288  11948 176534   3969 ,  250047  6169 641312  ,  531441 6561  2759 60819   1024 , 32768  391 7564  , 25 125  19849 1787743  ,  729000 8100  4537 305425  , 216 36  47 9191   256 , 4096



.816, 23309/

2985362173625  , 4096 262144  6142 480700  , 81 729

 207331217

.406, 8181/

P C

 1541761

1822623039  , 153664 60236288  11948 73513   3969 ,  250047  6169 109871  , 6561 531441  2759 93587   1024 ,  32768  391 7689  ,  125 25  19849 1058743  , 8100 729000  4537 305641  ,  216 36  47   256 ,  132871 4096

.816, 23310/

 207331217

,  2985362435769 4096 262144  6142 481429  ,  81 729

2, 1, 1

13 804  , 49 343  848961 782275425  ,  175616 3136  33 81  ,  64 16  4 62  ,  27 9

2, 1, 2

.93, 897/

.93, 896/

2, 2, 1 2, 2, 4 2, 1, 0

continued on next page



.406, 8180/

  13 ,  1147 49 343  848961 782099809  , 175616 3136  33 17  , 16 64  4 35  , 9 27 





162

Chapter 11 S-integer solutions of Weierstrass equations

continued from previous page

PC

m1 , m2 , m3

 , 132 49 343

 43

2, 0, 1 2, 0, 0 2, 0, 1 2, 0, 2 2, 1, 1

.14, 52/

33 

 

2, 1, 0 2, 1, 1

P C  43 475  ,  343 49

 74 ,  8

, 209 / . 106 49 343 ,  329  69 25 125  25 64  , 9 27  24 18  , 25 125



1, 1, 0

.3, 0/

2, 2, 0 2, 3, 1 1, 3, 0 1, 2, 0 1, 2, 1 1, 2, 2

1, 1, 1 1, 1, 2 1, 0, 2





,

,  552 343

33304 5642994  , 15625 1953125  221 2624  , 49 343  8159 390925   2916 ,  157464  1343 36575   576 , 13824  26 101  ,  125 25  25 111  , 8 4  66 252   25 , 125  31 116  , 9 27

.3, 1/ .4, 7/

 17 



 ,  39 16 64 151 1845  ,  64 512

1, 0, 1

.8, 21/

.8, 22/

1, 0, 0

.1, 1/

.1, 0/

1, 0, 1 1, 0, 2



.3, 4/



26 28  , 9 27

continued on next page





25 

16 64 1333  ,  151 64 512



204   69 , 25 125  25 91  ,  27 9  24 143  ,  125 25  625 14839  , 512 64

.4, 6/

 17

 74 , 25 8

49



 625

1, 1, 1

2, 1, 3

.14, 51/

 106

 ,  15351 64 512  33304 7596119   15625 ,  1953125  221 2967  ,  343 49  8159 233461   2916 , 157464  1343 50399   576 ,  13824  26 24  ,  125 25  25 119  , 8 4  66 377   25 ,  125  31 143  ,  27 9

2, 1, 2





.3, 3/



26 55  ,  9 27

163

Section 11.4 Example

continued from previous page

PC

m1 , m2 , m3

P C  58 559  , 81 729 9 7 , 4 8

 ,  1288 81 729  9 15  , 8 4

 58

1, 1, 2 1, 1, 1 1, 1, 0

.2, 3/

.2, 4/

1, 1, 1

.11, 35/

.11, 36/

.52, 374/

.52, 375/

1, 2, 1 1, 2, 0 1, 2, 1 1, 3, 3 1, 3, 1 1, 3, 1 0, 3, 2 0, 2, 1

 ,  133 16 64  172 1079  ,  729 81  338776 197184149  ,  3375 225  1219 9828  ,  625 15625  13961 1648791  , 100 1000  1793 68991  , 256 4096  68 1079   49 , 343

 , 69 16 64  172 350  , 81 729  338776 197180774  , 3375 225  1219 5797  ,  625 15625  13961 1649791  ,  100 1000  1793 73087  ,  4096 256  68 1422   49 ,  343

.21, 95/

.21, 96/

9

0, 2, 0

9

0, 2, 2

9 27  114 720  ,  343 49

 , 17 9 27  114 377  , 49 343

0, 1, 2

.342, 6324/

.342, 6325/

7

0, 2, 1

,

44 

7

21  ,  4 8

1

1

0, 1, 1

, 13 4 8



0, 1, 0

.2, 1/

.2, 0/

0, 1, 1

.1, 3/

.1, 4/

0, 1, 2 0, 0, 3 0, 0, 2 0, 0, 1



.37, 224/

  1525 ,  438957 2401 117649  49 32  ,  125 25 .0, 2/



.37, 225/

 1525 , 321308 2401 117649  49 93  ,  125 25 .0, 3/



List of symbols

On our compiling of the list we adopted the following rules.1 Mainly (but not exclusively) symbols introduced in this book are listed. For the definition or explanation of a symbol, the second column gives reference to either of the following: Definition (Def.), Fact (Fact), Lemma (Lemma), page (p.), Proposition (Prop.), Relation (Eq.), Section (Sec.), Theorem (Thm.). Symbols are listed by chapters. In each chapter, only the first occurrence of the symbol, where this is defined, is listed, except if its meaning is specialised subsequently. A symbol common to at least two chapters is listed in the chapter of its first occurrence, except if its meaning/context changes in a subsequent chapter. In the third column the meaning of the symbol very briefly and/or the context in which the symbol is used. Symbol

Reference

Meaning and/or context

Chapter 1

Elliptic Curves and Equations

gD0 Uij , Vij

Eq. (1.2) Def. 1.1.1

C,E

p. 4

X , Y, U, V

Eqs. (1.3), (1.4)

PC,PE .x.P /, y.P // .u.P /, v.P // a1 , a2 , a3 , a4 , a6 Etors r P1 , : : : , Pr P T

p. 4 ff.

homogenisation of g D 0 birational transformation between curves Ci , Cj birationally equivalent models usually representing elliptic curve with E short Weierstrass model birational transformation between curves E, C representatives of point P on models C , E coordinates of point P on model E coordinates of point P on model C coefficients of Weierstrass equation torsion subgroup of E rank of elliptic curve Mordell–Weil basis generic point of elliptic curve generic torsion point of elliptic curve

1

Eq. (1.6) Thm. 1.2.1

Eq. (1.7)

We were not able, however, to be absolutely consistent with them.

166

Symbol p p eL=K .P/ fL=K .P/ dL=K .P/ j  jp NL=K ./ HK ./ HK,fin ./ HK,1 ./ HK,fin .P / H./ h./ h.P / hD .P / O / h.P hOD .P / H

, , , , 

List of symbols

Reference

Meaning and/or context

Chapter 2

Heights

Sec. 2.1 p. 17 Eq. (2.1) above Eq. (2.3) Eq. (2.7) Eq. (2.9) Eqs. (2.4), (2.5) Sec. 2.3

exponent of prime ideal p exponent of rational prime p ramification index of prime ideal P degree of prime ideal P local degree of L=K at prime ideal P p-adic absolute value ideal norm with respect of L=K K-height of projective point finite-prime factor of K-height infinite-prime factor of K-height finite-prime height of polynomial P absolute height of projective point absolute logarithmic height naive or Weil height of point P height of point P used by S. David canonical height of point P canonical height used by S. David height pairing matrix least eigenvalue of H difference of naive from canonical height

Sec. 2.4 Def. 2.3.1 Sec. 2.5 Sec. 2.6 Prop. 2.6.1 Eq. (2.39) Prop. 2.6.2 Prop. 2.6.3 Chapter 3

Short Weierstrass equation over C

.!1 , !2 /

Sec. 3.1

g2 , g3 e1 , e2 , e3  A, B g2 , g3

Eq. (3.6) Fact 3.1.2 below Fact 3.1.2 Sec. 3.2



Fact 3.2.1

fundamental pair of periods for the Weierstrass }-function parameters of the Weierstrass }-function 4ei3  g2 ei  g3 D 0 (i D 1, 2, 3) discriminant of the Weierstrass }-function D !2 =!1 coefficients of Weierstrass Eq. over C parameters of the Weierstrass }-function g2 D 4A, g3 D 4B group isomorphism from a fundamental parallelogram to E.C/ isomorphism : E.C/ ! C=ƒ; inverse of  above

Sec. 3.3

167

List of symbols

Symbol

Reference

Meaning and/or context

Def. 3.3.2

"P

Sec. 3.4.4 Sec. 3.5 Thm. 3.5.1

l0 l

Thm. 3.5.2 Eq. (3.31)

elliptic logarithm; as above with values on a specified fundamental parallelogram least positive real period of } when g2 , g3 2 R period 2 i R with least positive imaginary part, when g2 , g3 2 R fundamental pair of periods when g2 , g3 2 R, >0 fundamental pair of periods when g2 , g3 2 R,